Bob, I've been approaching my NY style two different ways- a blistery relatively high char 3 minute pie with a pre-heat of 600 for an hour, or, a more evenly browned 4 minute pie with a pre-heat of 500. The first is kind of an 'adult' pie, where the second is more for less sophisticated eaters.
Hotsawce, I've done punch downs straight out of the fridge (and in the fridge, mid-ferment), and they've produced pretty good results, but this was with doughs with hydrations of over 68%. When dough is cold, gluten is in a more rigid/fragile state. With an elevated hydration/slack dough, a punch down doesn't tear the gluten, but with a more traditional lower level of hydration, I think it's a bit too aggressive.
Coming from a baking background I was pretty pro punch downs (slap happy?
), doing as many as 4 prior to skinning, but, now that I look back, I see that as being overkill. One punch down is more than enough to redistribute the nutrients for the yeast and whip them into a frenzy.
When I resolve my container issues and go back to punch downs, I'm probably going to settle in on something like this:
Remove dough from fridge and allow to come to room temp (with my plastic containers, that's about 2.5 hours)
Allow dough to rise to pre-punch down volume (probably about 2 hours)
For my 65% hydration bromated All Trumps dough, that translates into 3x volume coming out of the fridge, 4x at room temp, about 2x after punch down and then back to 4x for the final rise. AT @65% (and higher) seems to allow for crazy volumes. I've taken it to as many as 9x without any adverse effects. I also, with my cool water dough and straight to fridge technique, get a pretty large fermentation window coming out of the fridge. I'm able to hold my 3 day cold fermented dough at room temp (70) for as long as 10 hours without any signs of overfermentation. If there's any chance you've got an overfermented dough coming out of the fridge, skip the punch down.
Cold ferments develop a lot of gluten, so I tend to be very cautious with my kneading (not quite smooth). If you're knead happy on the front end and potentially looking at fragile overworked gluten after the cold ferment, then a punch down is the last thing you want to do.