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 watch for golden brown specks to form at the bottom and a nutty aroma to develop, about 5–7 minutes. Remove from heat immediately and let cool slightly.
03 - In a medium bowl, whisk together the 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. Add eggs one at a time, mixing thoroughly after each addition. Stir in vanilla extract, mashed bananas, sour cream, and whole milk until fully blended.
05 - Gradually add the dry ingredient mixture to the wet ingredients. Fold gently with a spatula until just combined — avoid overmixing to keep the cake tender.
06 - Pour the batter into the prepared pan and spread evenly. Bake for 35–40 minutes, or until a toothpick inserted into the center comes out clean with no wet batter.
07 - Allow the cake to cool completely in the pan set on a wire rack. Do not frost while warm or the frosting will melt.
08 - Beat the softened cream cheese and butter together using a hand mixer or stand mixer until creamy and lump-free. Gradually add sifted powdered sugar, vanilla extract, and salt. Continue beating on medium-high speed until the frosting is thick, fluffy, and spreadable.
09 - Spread the frosting generously and evenly over the completely cooled cake. For cleaner slices, chill the frosted cake in the refrigerator for 20–30 minutes before cutting into squares.