01 - Preheat the oven to 350°F. Line a 9x13-inch baking pan with parchment paper and lightly grease.
02 - In a large bowl, whisk together melted butter and sugar until smooth. Add eggs one at a time, mixing well after each, then stir in mashed bananas and vanilla extract.
03 - In a separate bowl, combine flour, baking powder, and salt. Fold the dry ingredients into the wet mixture until just combined—do not overmix.
04 - Spread batter evenly into the prepared pan. Bake for 27-32 minutes, or until a toothpick inserted in the center comes out clean. Let cool completely in the pan.
05 - For the frosting: In a saucepan over medium heat, cook butter until it turns golden brown and smells nutty, about 4-5 minutes. Remove from heat and let cool slightly for 5 minutes.
06 - In a mixing bowl, combine the browned butter, powdered sugar, 2 tbsp milk, vanilla extract, and salt. Beat until the frosting is smooth and spreadable, adding up to 1 more tablespoon of milk if needed.
07 - Spread frosting over cooled brownies. Cut into squares and serve.