01 - Preheat oven to 350°F. Line a 9x13-inch baking pan with parchment paper, allowing overhang on sides for easy removal. Lightly grease the parchment with butter or cooking spray.
02 - In a large mixing bowl, whisk melted butter and granulated sugar together until completely smooth and sugar begins to dissolve, about 2 minutes.
03 - Add eggs one at a time, beating thoroughly after each addition. Stir in mashed bananas and vanilla extract until fully incorporated.
04 - In a separate medium bowl, whisk together flour, baking powder, and salt until well blended.
05 - Gently fold dry ingredients into wet mixture just until combined—do not overmix. Spread batter evenly into prepared pan using an offset spatula.
06 - Bake for 27-32 minutes until toothpick inserted in center comes out clean or with few moist crumbs. Cool completely in pan on wire rack, approximately 1 hour.
07 - Place butter in light-colored saucepan over medium heat. Cook, swirling pan occasionally, until butter foams, turns golden brown, and emits nutty aroma, approximately 4-5 minutes. Remove from heat and let cool 5 minutes.
08 - Combine browned butter, powdered sugar, 2 tablespoons milk, vanilla, and salt in mixing bowl. Beat with electric mixer until smooth and spreadable. Add additional milk, 1 teaspoon at a time, if needed to reach desired consistency.
09 - Spread frosting evenly over completely cooled brownies. Lift from pan using parchment overhang and cut into 16 squares. Serve at room temperature.