Listen to Jesse.
Do not make banana pudding with green bananas. Even if the fruit has a tinge of green on the ribs, the banana will be hard and bitter. You must use bananas that are ripe.
Now, you’re not going to find ripe bananas in the grocery store–sometimes I think the public has been conditioned by years of buying green bananas that any banana with a dark spot is spurned–so you have to buy them a bit green.
Fortunately, bananas are climacteric; they ripen after picking. My daddy, who was stationed in the Pacific during WWII, knew quite well how a banana tasted, and put bunches on top of the refrigerator–where we couldn’t reach them–to ripen.
Place grocery bananas in a paper sack–plastic doesn’t let the fruit breathe–and give it a good twist to keep out fruit flies. In two or three days, when the fruit is soft and aromatic you’re ready to make banana pudding. Or banana pound cake. Or peel and eat.