FWIW, the broken banana thing is age-appropriate for toddlers and a typical developmental thing. I read an explanation of it once, though it sounded a bit Freudian. Anyway, lots of kids do this. I think it is sometimes worth it to just let them cope with the broken banana.
At that age with my explosive toddler (my older son)...I used to eat the last banana at night if we only had one left. If we only had one left and he asked for it and the damn thing broke then there wasn't another one to try...he expected me to drop everything and go to the store for another bunch of bananas. If I ate the single left banana before he could ask for it, we were not in that position. I even stopped getting bananas for a while and switched to different fruit until his flexibility on bananas was better...and for a while I told him he could only have them sliced up with a dollop of whipped cream or yogurt (thus avoiding the whole "it broke so now it is no good" problem).
