Butter a 9x9-inch baking dish with 2 tablespoons of softened butter using your hands or a pastry brush.
Line the bottom and sides of the baking dish with a single layer of vanilla wafers.
Arrange a single layer of sliced bananas over the vanilla wafers.
Beat cream cheese and vanilla extract with a mixer until smooth, about 1 to 1½ minutes.
Add powdered sugar and 1 cup of whipped topping to the cream cheese mixture and beat until smooth, about 1½ to 2 minutes.
Spread the cream cheese layer evenly over the banana and wafer layer.
Add another single layer of vanilla wafers over the cream cheese layer, then top with the remaining sliced bananas.
In a mixing bowl, combine cold milk, 1 cup whipped topping, and pudding mix. Stir until thickened.
Spread the pudding layer evenly over the banana and wafer layer.
Cover the dish and chill in the refrigerator for at least 2 hours.
Before serving, spread remaining whipped topping over the pudding layer and sprinkle with crushed vanilla wafer crumbs.
Slice into 9 pieces and serve chilled.