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 - Melt butter in a medium saucepan over medium heat. Stir continuously as it foams and watch for the milk solids to turn golden brown and release a nutty aroma, about 5 to 7 minutes. Remove from heat and let cool slightly.
03 - In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until evenly distributed.
04 - In a large mixing bowl, combine the browned butter with granulated sugar and brown sugar. Beat until smooth and well incorporated.
05 - Add eggs one at a time, mixing thoroughly after each addition. Stir in vanilla extract, mashed bananas, sour cream, and whole milk until combined.
06 - Gradually add the dry ingredient mixture to the wet ingredients, folding gently until just combined. Avoid overmixing to keep the crumb tender.
07 - Pour batter into the prepared pan, spreading evenly. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
08 - Transfer pan to a wire rack and let the cake cool completely before frosting.
09 - Beat softened cream cheese and butter together in a bowl until creamy and lump-free. Gradually add sifted powdered sugar, vanilla extract, and salt. Continue beating until the frosting is thick, fluffy, and spreadable.
10 - Spread frosting generously and evenly over the cooled cake. For cleaner slices, chill briefly before cutting into squares.