Preheat and Prepare: Preheat your oven to 350°F (175°C). Grease a 9-inch square baking pan lightly so that the brownies release easily.
Make the Brownie Base: In a large bowl, combine the melted butter, granulated sugar, eggs, and vanilla extract. Whisk these together until the mixture is smooth. Then, add the cocoa powder, all-purpose flour, salt, and baking powder. Stir gently until just combined, being careful not to overmix. Transfer the batter into your pre-greased baking dish, then use a spatula to evenly level the surface.
Bake the Brownies: Place the pan in the preheated oven and bake for about 18 to 20 minutes. Check the center with a toothpick; it should come out with a few moist crumbs. Once done, remove the pan from the oven and allow the brownies to cool slightly.
Prepare the Banana Pudding Layer: While the brownies are cooling, combine the mashed bananas, granulated sugar, cornstarch, milk, vanilla extract, and a pinch of salt in a small saucepan. Set your saucepan on medium-low heat and keep stirring continuously. As you stir, the mixture will gradually thicken and begin to bubble gently, which should take about 5 to 7 minutes. Once thickened, remove the pudding from the heat and let it cool for a few minutes.
Assemble the Dessert: Once the brownie base has cooled a bit, pour the banana pudding evenly over the top. Spread the pudding gently to cover the surface of the brownies.
Chill and Serve: Place the assembled dessert in the refrigerator for at least 30 minutes so that the layers set together nicely. After chilling, cut the dessert into 12 squares and serve.