Preheat oven to 350°F (175°C). Grease or line a 9x13-inch baking pan with parchment paper.
In a large bowl, whisk together melted butter and brown sugar until smooth.
Add eggs and vanilla extract; whisk until combined.
Stir in mashed bananas.
In a separate bowl, whisk together flour, baking powder, and salt.
Gradually add dry ingredients to wet ingredients and mix until just combined.
Spread batter evenly into prepared pan.
Bake for 25–30 minutes or until a toothpick inserted in the center comes out with a few moist crumbs.
Cool completely.
In a medium bowl, whisk banana pudding mix with cold milk for 2 minutes. Let thicken for 5 minutes.
Fold in whipped topping until smooth.
Spread pudding mixture evenly over cooled brownies.
Sprinkle crushed vanilla wafers over the top.
Add additional whipped topping and banana slices if desired.
Refrigerate for at least 1 hour before slicing and serving.