01 - Preheat the oven to 350°F. Line a 9x13-inch baking pan with parchment paper and lightly grease the paper and sides of the pan.
02 - In a large bowl, whisk together melted butter and granulated sugar until smooth and well combined.
03 - Add eggs one at a time, mixing thoroughly after each addition. Stir in the mashed bananas and vanilla extract until fully incorporated.
04 - In a separate medium bowl, whisk together flour, baking powder, and salt until evenly distributed.
05 - Gently fold the dry ingredients into the wet mixture until just combined. Be careful not to overmix—the batter should remain thick and moist.
06 - Spread the batter evenly into the prepared pan. Bake for 27–32 minutes, or until a toothpick inserted in the center comes out clean or with just a few moist crumbs. Allow to cool completely in the pan.
07 - Place the butter for the frosting in a saucepan over medium heat. Cook, swirling the pan occasionally, until the butter foams, turns golden brown, and develops a nutty aroma, about 4–5 minutes. Remove from heat and let cool for 5 minutes.
08 - In a mixing bowl, combine the cooled browned butter, powdered sugar, 2 tablespoons milk, vanilla extract, and salt. Beat with a mixer or whisk until smooth and spreadable. Add up to 1 additional tablespoon of milk if needed to reach desired consistency.
09 - Spread the frosting evenly over the completely cooled brownies. Cut into 16 squares and serve.