Even after they've cooled, I still break them down or shred. It just takes a little more effort than when they are warm. I usually find by the time I'm done cooking and cleaning up, the last thing I want to do is shred/break down the leftovers from a carcass. So I just usually wrap in foil and deal with it later. Then I pull/shred it and use it for chicken salad, nachos, quesadillas, etc.