01 - Preheat oven to 350°F. Grease a 9x13 inch baking pan and line with parchment paper, leaving overhang on the sides for easy removal.
02 - Place unsalted butter in a medium saucepan over medium heat. Stir continuously as it melts, foams, and eventually develops a golden brown color with a fragrant nutty aroma, approximately 5 to 7 minutes. Remove from heat and allow to cool slightly.
03 - In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until evenly distributed. Set aside.
04 - In a large mixing bowl, combine the browned butter, granulated sugar, and brown sugar. Beat until well incorporated. Add the eggs one at a time, mixing thoroughly after each addition. Stir in the vanilla extract, mashed bananas, sour cream, and whole milk until smooth.
05 - Gradually add the dry ingredient mixture to the wet ingredients, folding gently until just combined. Be careful not to overmix to keep the cake tender.
06 - Pour the batter into the prepared pan and spread evenly. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
07 - Transfer the pan to a wire rack and let the cake cool completely before frosting.
08 - In a mixing bowl, beat the softened cream cheese and butter together until creamy and smooth. Gradually add the sifted powdered sugar, vanilla extract, and salt. Continue beating until the frosting is thick, fluffy, and spreadable.
09 - Spread the frosting generously and evenly over the cooled cake. For cleaner slices, chill the cake briefly before cutting into squares.