01 - Preheat oven to 350°F. Line a 9x13-inch baking pan with parchment paper and lightly grease with butter or cooking spray.
02 - In a large bowl, whisk melted butter and granulated sugar until smooth. Add eggs one at a time, mixing well after each addition. Stir in mashed bananas and vanilla extract until fully incorporated.
03 - In a separate bowl, whisk together all-purpose flour, baking powder, and salt until evenly distributed.
04 - Gently fold the dry ingredients into the wet mixture using a spatula. Mix until just combined—do not overmix or the brownies will become tough.
05 - Spread batter evenly into the prepared pan. Bake for 27–32 minutes until a toothpick inserted in the center comes out clean or with just a few moist crumbs. Let cool completely in the pan before frosting.
06 - In a saucepan over medium heat, cook the butter, swirling occasionally, until it turns golden brown and releases a nutty aroma, about 4–5 minutes. Remove from heat and let cool for 5 minutes.
07 - In a mixing bowl, combine 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 more tablespoon milk if the frosting is too thick.
08 - Spread frosting evenly over the completely cooled brownies. Cut into 16 squares and serve. Store in an airtight container at room temperature for up to 3 days.