Place the whole banana in its skin on a parchment-paper lined sheet, and roast in the oven at 400° F for 15 minutes or until the skin blackens.
Remove the roasted banana from skin and lightly mash with a fork. Set aside.
Place the milk, sugar, and cornstarch in a saucepan, mix well with a whisk, and heat on medium, continuously stirring. When liquid comes to a boil, let it boil for 30 seconds, add the mashed banana, and then turn down the heat to low and continue to stir until mixture is thick.
Beat the egg in a separate dish with a couple of tablespoons of pudding, and then incorporate egg mixture into pudding. Mix in the vanilla and butter. Remove pudding from heat.
You can either make individual pudding cups or one big pudding dish, but here’s the general procedure for assembly: layer some of the sliced bananas and cookies in a dish, and then pour some pudding over bananas and cookies until dish is half full. Repeat layers and then stick some of the cookies on top in a decorative flourish.