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 the unsalted butter in a medium saucepan over medium heat. Stir continuously as it foams, watching carefully, until it turns a golden amber color and releases a nutty aroma, about 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.
04 - In a large mixing bowl, combine the browned butter with granulated sugar and brown sugar. Beat until well incorporated. Add eggs one at a time, mixing thoroughly after each addition. Stir in 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. Take care not to overmix to keep the cake tender.
06 - Pour the batter into the prepared pan, spreading it 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 - Using a hand mixer or stand mixer, beat the softened cream cheese and butter together until creamy and lump-free. Gradually add sifted powdered sugar, vanilla extract, and a pinch of salt. Continue beating until the frosting is thick, fluffy, and smooth.
09 - Spread the frosting generously and evenly over the cooled cake. For cleaner slices, chill the frosted cake briefly before cutting into squares. Store any leftovers covered in the refrigerator for up to 4 days.