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 continue cooking until it turns a golden amber 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 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 fully combined.
06 - Gradually add the dry ingredient mixture to the wet ingredients. Fold gently until just combined, being careful not to 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.
08 - Allow the cake to cool completely in the pan on a wire rack before frosting.
09 - Beat softened cream cheese and butter together until creamy and smooth. 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. Chill briefly for cleaner slicing if desired, then cut into squares and serve.