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 develops a golden brown color with a fragrant 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, granulated sugar, and brown sugar. Beat until 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 the mixture is smooth.
06 - Gradually fold the dry ingredients into the wet mixture. Stir until just combined — avoid overmixing to keep the cake tender.
07 - Pour the batter into the prepared pan and spread evenly. Bake for 35 to 40 minutes, or until a toothpick inserted in the center comes out clean.
08 - Allow the cake to cool completely in the pan set on a wire rack before applying frosting.
09 - Beat the softened cream cheese and butter together until creamy and smooth. Gradually add sifted powdered sugar, vanilla extract, and a pinch of salt. Continue beating until the frosting is thick and fluffy.
10 - Spread the frosting generously over the cooled cake in an even layer. Chill briefly for cleaner slicing if desired.