I think this is a move that most of us here tend to do, but I'd like to review it in a little more detail.
The move is this: You've bet out pot or raised pot with whatever holdings on the flop, then the board pairs on the turn. If I'm the one who had the initiative here, I'll half-pot it almost every time at that point. VERY occasionally, and only against a super-LAG, I'll check if I'm full.
Now, I see a lot of players checking their fulls (and should make a more prominent note of this) in this situation, inviting villain to catch a draw that he thinks might be good or to try to buy. That invitation also has some advantages, because they will bite some of the time (I've definitely done it as the one drawing--I make my nut flush and then call and lose when he pots the river). However, I think what you lose here are the bluffs when you were betting your straight draw in the first place (or betting top 2 and the bottom card paired, actually hurting your hand against an overpair). I really think the bluff-factor here more than compensates for not getting paid off as much by the draw.
But I have 2 exceptions that I also wonder about. The first exception is when I actually quadded up. I'll typically check that, and I'm not sure it's a good play.
The second exception (which objectively seems even worse) is when I don't have the full and the pot is just too big--like half pot is at least 75% of my remaining stack. This is usually in a raised pot where a couple of flop callers can already make for a very big pot in the first place. But it also happens, for example, if one guy bets, gets a caller or two, I raise in LP, and at least one player calls. That's also a pretty big pot.
Now, the vast majority of the time I raised there, I really did have a set, simply because these are more frequent than monster draws.
If I'm not full here, I just usually don't have the balls to make the bluff for the rest of my stack, so if I bet it there, I almost always really am going to be full.
I'll leave it at that for the moment as to the exceptions, which are really one of the big issues here--also if one really should make more or fewer exceptions.
Here's another thing that I've had happen: I make my half-pot bet and get a mini-raise (or occasionally a big raise). If I'm on the draw, I'll just give them credit for the full and fold rather than draw dead. So, that's another question: Should one really ALWAYS do that?
But let's say I really am full here. Ok, now I give them credit at least for a full house (possibly quads). How do you play the hand from there? There are also some definite issues here regarding the quality of your full house as well as what they're prepared to call/bet. This is also perhaps a weakness in not making the same move on my own quads--particularly in "normal" pots, where there's still a fair amount of stack-depth left.
Anyhow, in this regard, I think that the big raise is more likely to be a bluff or at best the worst possible full house--although if they give me credit for the full, it may also be quads sometimes (I think I've had that happen, not sure).
One possible answer to these possibilities: If you have the nuts (for example, you had top 2 and top card pairs), then go ahead and push. If you have a very low full (you had bottom set and one of the top cards pairs), you fold. And on the mediocre fulls you just call, according to the philosophy that they're likely to keep betting with worse hands but will only call with superior ones.
As an aside: I think this is one of the reasons that I rarely make these post-flop posts--namely, while this move is fairly simple, there are definitely LOTS of "ifs," "ands" and "buts." Like, maybe if the pot is more multi-way and stacks shallow relative to pot-size, you should be more inclined to check your overfull or quads--it's at least more likely that someone else is also full.
Ok, enough for now. I'll be interested to see if we can make some headway on the more complex aspects of this play.