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 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 immediately and let 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 light brown sugar. Beat until smooth and well incorporated.
05 - Add the eggs one at a time, beating after each addition until fully incorporated. Stir in the vanilla extract, mashed bananas, sour cream, and whole milk. Mix until smooth.
06 - Gradually add the dry ingredient mixture to the wet ingredients, folding gently with a spatula. Mix until just combined and no dry streaks remain. Do not overmix.
07 - Pour the batter evenly into the prepared pan. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean and the top is golden.
08 - Allow the cake to cool completely in the pan set on a wire rack. Do not frost while warm.
09 - Beat the softened cream cheese and softened butter together in a bowl until creamy and lump-free. Gradually add the sifted powdered sugar, vanilla extract, and salt. Continue beating until the frosting is thick, fluffy, and spreadable.
10 - Spread the cream cheese frosting generously and evenly over the completely cooled cake. Chill briefly in the refrigerator for cleaner slicing if desired.