01 - Preheat oven to 350°F. Line a 9×13-inch baking pan with parchment paper and lightly grease the surface.
02 - In a large bowl, whisk together 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, combine all-purpose flour, baking powder, and salt. Stir to distribute evenly.
04 - Gently fold the dry ingredients into the wet mixture until just combined. Do not overmix—small lumps are acceptable.
05 - Spread batter evenly into the prepared pan. Bake for 27 to 32 minutes, or until a toothpick inserted in the center comes out clean. Let cool completely in the pan before frosting.
06 - In a saucepan over medium heat, cook butter until it turns golden brown and smells nutty, about 4 to 5 minutes. Remove from heat and let cool slightly for 5 minutes.
07 - In a mixing bowl, combine the browned butter, powdered sugar, 2 tablespoons milk, vanilla extract, and salt. Beat until smooth and spreadable, adding up to 1 more tablespoon of milk if needed to achieve desired consistency.
08 - Spread frosting evenly over cooled brownies. Cut into squares and serve.