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 butter in a medium saucepan over medium heat. Stir continuously as it foams and develops a golden brown color with a 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.
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.
05 - Stir the vanilla extract, mashed bananas, sour cream, and whole milk into the wet mixture until smooth and homogeneous.
06 - Gradually fold the dry ingredients into the wet mixture, stirring until just combined. Avoid overmixing to keep the cake tender.
07 - Pour the batter evenly into the prepared baking pan. Bake for 35 to 40 minutes, or until a toothpick inserted into the center emerges clean.
08 - Transfer the pan to a wire rack and let the cake cool completely in the pan before frosting.
09 - Beat the softened cream cheese and butter together until creamy and smooth. Gradually add the sifted powdered sugar, vanilla extract, and salt, continuing to beat until the frosting is thick and fluffy.
10 - Spread the frosting generously over the cooled cake in an even layer. Chill briefly for easier slicing if desired, then cut into squares and serve.