(tl;dr Fold flop)

I was first thinking about looking at flop-turn-river continuation ranges and bluffcatching, but it can be done much simpler. But I've not posted rambles in a while, so I will make it long. 

Analysis assuming Villain has all AA in his 4B range:

1. You are 35% against his range (putting him on 3%6h) on the flop
2. What's worse, you have very little potential to outdraw the current nuts and you don't have future blocker value

You're setting yourself up for bluffcatching without outs over 3 streets OOP. These situations tend to not work out well, especially when Villain's range is strong. Even if you knew Villain would bet his range on all streets, you could not call down:

EV (calldown vs range)
= 0.35(81 + 2*495) - 495
= -€120

So to start calling, you need him to stop betting at some point and let you show down. And/or you must find bluffing opportunities. This brings us to the third problem:

3. Villain has flopped the nuts 66% of the time. The board will change from flop to river, but his range is so strong that it's probably correct (at least not exploitable) for him to bet all his air on flop and turn, and then he can play the river with a huge range/information/position advantage in a big pot. Oi, that's bad.

Let's talk about that bluff-his-range-on-flop-and-turn-concept a bit:

Why it's correct to bluff more early than late (toy game example)

Whether Villain can barrel all his air depends on how often he has the nuts (or in practice, the effective nuts) on the river. 

Let's use a nuts/air toy example where Villain has 100 hands on the flop and 40 of them are nut hands (and they stay that way). On the river he can pot all of these and add another 40/2 = 20 as bluffs, betting 60 total.

Since Hero is indifferent on the river when Villain is balanced, he might as well fold. But that means Villain's 60 river bets are effectively nuts on the turn. So he can pot turn and add another 60/2 = 30 hands as a bluffs there. Betting 90 hands on the turn, 40 value and 50 bluffs. 

Using the same logic and going backwards to the flop, Villain can add another 90/2 = 45 bluffs, and he actually ends up not having enough bluffs to use there. Because his range is so extremely strong, having the nuts on 40% of rivers.

Now, we haven't proved conclusively that Villain actually has the nuts often enough by the river to bluff 100% on flop and turn, but it seems pretty damn likely. He has 66% nuts on the flop, we have 0% nuts, and he knows we have a bluffcatcher (with or without some outs) once we start calling. Even if many turn/river runouts leaves him without the nuts, he will usually know where he's at when we keep checking. He will have the effective nuts very often.

What should Villain do?

With such a super strong range, it seems obvious that Villain maximizes his EV by betting range on flop and again on turn (if not all turns, then most of them), building a big pot against your bluffcatcher, and then making you indifferent on the river. Then you lose even more than the -€120 from calling down vs his range, since some of his bluffs stop betting.

This is somewhat compensated for when the board runs out so  bad for him that his top set stops valuebetting, but his positional advantage + information about your range will allow him to bet a lot without the absolute nuts. Regardless, you won't do well.

If he has a read that you call or fold too much, he can exploit you by shutting down his bluffs entirely on the river, or bluffing everything. Those are costly mistakes for you to make in big pots, and when this happens, your EV drops further.

There are range considerations, too (hero's flop check-calling range has hands with decent chance to improve to future nuts + blocker hands that can rep it), but I leave that for another poster.

Conclusion

Calling down against his range is -€120. Villain can do better by being balanced on the last bet (or by exploiting you). We don't have outs. And we have much better hands than this in our range to check-call and valuebet/bluff later (hands with outs/blockers that can make us money on future streets). So we conclude:

You must fold on the flop.

Yes, fold. It's a freak flop for you, and don't worry about it.

Afterthought

We could change the composition of his 4B range, too. If he flats the 3-bet with poor AA, things are looking better for us, at least on the flop.

Yeah, I agree with ZF here.  It sounds ugly, but there aren't much we can do with our dry TT here.

We talked a bit about this spot and I had an idea of x/c'ing the flop to turn our hand into a bluff on KQJ turn, but that bluff has to work insanely often to be profitable, so I just discarded that idea.

However we learned that it's worth mentioning a few other points here, which I think are more interesting and important points than the original hand per se.  I'm off to climbing now, but if I still feel motivated after coming back, I'll post my thoughts on this.  Otherwise, I'll try this again tomorrow :)

Climbed for 2 hours, I'm beyond exhausted.  I'll come back with analysis tomorrow..

Alright, here are my thoughts on our range play in this spot.

Should we call on flop?

ZF made an excellent post on this, so I will just point to the cliff.  Villain has top set on this board about 65-70% of the time, and we can't.  We might think, he can be exploiting us by bluffing often, but there isn't much we can do about it.  Why? When villain has the nut ~70% of the time, we simply can't call down even if he is ALWAYS bluffing with all the other hands. 

Can we float on flop then?

At this point, it occurred to me: what if we x/c here with the intention of bluffing on some turn/rivers, mostly K/Q/J?  At first, the idea seemed kinda cool for a couple of reasons.  When we x/c on this flop in a 4b pot, we don't really have much air, and a huge part of our range should be sets, 2p, and pair + KQJ.  Thus, if we donk on the straightening turns, it is fair to assume that we will get some credit and won't get raised unless he had some blockers or a straight himself. 

According to PPT, it turns out that on K/Q/J turn, his 3% range should have a straight less than 9% of the time.  Thus, I thought we could pot on this turn, and if he just flats, we can bomb again on the river to get a fold.

However, a more careful observation revealed a different story.  On these turns, he can have blockers (for example, KK/QQ on J turn) about 26% of the time.  And if the turn is a non-heart, it brings a flush draw and he will have AA+bdf about 15-20% (on average) of the time.  

Granted, he was not gonna fold his top set to our turn pot bet anyway, and that's fine.  But if he has top set + something to go with, not only is he continuing on the turn 100% of the time, but he will improve to a better hand (boats, quads, flush) on the river a significant amount of time!  On As2hTd Js board, for example, he will improve to a better hand ~24% of the time, and we simply can't bluff that often.  Fwiw, for our flop float + turn bluff plan to be profitable, our bluff has to work close to 70% of the time.  For, if we assume that we are gonna pot on those turn and jam on blank rivers, we invest 40 on flop, 120 on turn, and 360 on the river = 520 in total, to win 120 (flop) + 120 (turn) = 240.  Not to mention that on non blank turns, we mostly have to check/fold.  Turning our flopped middle set into a bluff sounds nice and dandy, but it's just not gonna work this often.

Even without calculating all this, there is another way to look at it, which ZF has mentioned.  We have a hand with no blocker and very little (weak) backdoor draws.  If we were able to profitably bluff with this hand, that means we can get away with bluffing in this spot with ~100% of our 4b flatting range, and intuitively that just can't be true, especially in spots like this where our ranges (his and ours) are significantly asymmetric.

This brings us to the next point. 

With what hands can we float on the flop and (possibly) bluff later?

The easiest answer would be: hands with blockers.  Now, what blockers?

Intuitively, Axxx sounds like a good hand to bluff with.  When we hold the top set blocker, he can have AA "only" 57% of the time, as opposed to 66-70% when we don't block it.  This difference is somewhat significant, but not huge enough that we can get away with going crazy.  Truth be told, he can still have nuts quite often, and we can't.  In huge pots, this can be a difference between life and death.

Now, what about straight blockers, namely KK/QQ/JJ?  When we call his flop c-bet, it is reasonable to assume that we will often have a pair to go with it, so let's narrow it down to {AKKx, AQQx, AJJx}$ss and {KKQT, KKJT, KQQT, QQJT, KJJT, QJJT}$ds.  

As expected, it turns out these hands have significant blocker effects.  When we have KJJT$ds on As2hTd Qs board, for example, he can have blockers 14% of the time (compared to 26% when we don't block them), and a straight less than 1% of the time.  Interestingly enough, the (KK,QQ,JJ)!A part of our range doesn't block his backdoor FD very much, whereas (AKK, AQQ, AJJ) part does; this is due to the fact that he can have tons of NFD with his AA.  In this sense, the best turn cards for us to bluff on would be K/Q/J of spades, because he can't have NFD as nearly often on this cards.

Now, how effective is our bluffing gonna be?  All I can tell is, it's definitively more effective than when we don't hold a blocker, but probably not enough to show a huge profit, if at all.  The calculation is gonna be somewhat complicated and will be omitted here; but it is worth mentioning that, when we x/c on the flop with this range, not only will we have some bluffing opportunities that will arise, but sometimes we can also improve significantly on the turn!  With KKQT$ds for example, we can sometimes turn the nuts (~10% of the time), backdoor FD (~40% of the time), as well as "turn a blocker" (~20% of the time), etc.  

Thus, I would be much more inclined to bluff with these hands.  Still, it's gonna be an uphill battle.  Playing OOP in a 4-bet pot with a capped range vs nutty range just doesn't sound like a spot where we can print much money.  However, he bet small on the flop and we might have enough equity (in a broader sense), so my gut tells me x/c'ing cannot be that bad.  I can be convinced otherwise, though.

Now, this inevitably throws the following question at us:

Would we need to donk with our turned straights?

Yes, I think so.  First off, K/Q/J improve our range significantly; if villain had a top set, these are the worst cards for him and he can easily check back for pot control/deception/whatnot.  At the same time, he won't just fold to our single pot bet on the turn; he might think he has implied odds for boating up (which he shouldn't), flushing up/bluffing on flushing cards, etc.  Thus, rather than risking him checking back on the turn and freely realising his equity in a huge pot, we might as well pot into him and get the maximum value.  Of course, we were gonna bluff on these turn cards anyway, so we should be value betting sometimes.  

There might be some exceptions depending on his postflop tendencies, and we might argue that we can go for a x/r on K/Q/J of clubs that don't bring a FD.  However, as a general plan I would like to see myself donking on these turns with an actual straight.

That said, there is one more important scenario that we should consider imo:

What if we had AKQJ$ds?

AKQJ$ds are only a few combos (was gonna say a handful of combos, but realised it had more combos than my digits), but these are both interesting and important hands to analyse.  First of all, although this hand doesn't have a significant blocker effect (he can still have nuts 70% of the time), this is a hand (or a range of hands, to be exact) that we want to have in this spot against his nutty range.  

As cliche as this might sound, on K/Q/J we improve to the best hand, and we can pick up backdoor FD often enough.  In short, our visibility is excellent.  Compare this to when we have a middle set; he can still have AA about the same amount of time, and our hand can never improve to the best, nor pickup some other potentials, unless we hit our 1-outer.  

(I'd like to call AKQJ$ds a "live" or "clean" hand, and naked TT a "dead" or "dirty" hand.  I thought these are cool terms, and hope nobody copyrighted them before for obvious reasons!)

For our purpose, we can make a simple model where we x/c with AKQJ$ds and bet the turn/river ONLY for value, without bluffing or semi-bluffing.  My laptop is almost dying, so I will get back to this when I get home and/if the electricity is back on..