Preheat the oven to 350°F or 175°C and prepare a 9x9-inch baking pan with nonstick cooking spray and parchment paper. In a large mixing bowl, combine melted butter and brown sugar until well blended. Beat in the eggs one at a time, then stir in the vanilla extract.
In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Fold in the mashed bananas (and 1 cup chopped toasted walnuts if you want) until evenly distributed.
Spread the batter evenly into the prepared pan and bake for 35 to 45 minutes or until a toothpick inserted into the center comes out with a few moist crumbs.
While the base is baking, prepare the banana pudding layer by whisking together the instant banana pudding mix and milk. Or make our Pudding Frosting Let it set in the refrigerator until the base has cooled.
Once the base is cool, spread the banana pudding layer evenly on top. Refrigerate the brownies for at least 2 hours to allow the pudding layer to set.