Preheat oven to 350 degrees F (175 degrees C). Spray a 9-inch cake pan with cooking spray.
Combine brown sugar, half of the pudding mix, and cinnamon in a large resealable plastic bag. Seal and shake well until thoroughly combined and no lumps remain.
Pour the remaining half of the pudding mix and 1/2 cup melted butter into a small bowl. Stir until combined.
Dip each biscuit piece in the butter mixture and shake in the sugar mixture. Arrange half of the pieces into a layer on the bottom the prepared pan. Drop banana over the layer; sprinkle chocolate chips on top. Cover with a layer of of the remaining biscuit pieces.
Bake in the preheated oven until golden brown and no longer doughy in the center, 18 to 22 minutes. Cool monkey bread in the pan for 10 minutes. Invert onto a serving plate.
Beat confectioners' sugar, 2 tablespoons butter, and cream in a bowl until smooth. Drizzle over monkey bread.