01 - Preheat oven to 350°F. Grease a 9×13-inch baking pan and line with parchment paper, leaving overhang on the long sides for easy removal.
02 - Melt 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, toasted aroma, about 5–7 minutes. Remove from heat immediately and let cool for 10 minutes.
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 on medium speed until well incorporated, about 2 minutes.
05 - Add eggs one at a time, beating after each addition until fully incorporated. Mix in vanilla extract, mashed bananas, sour cream, and whole milk until smooth and combined.
06 - Gradually add the dry ingredient mixture to the wet ingredients, folding gently with a spatula or mixing on low speed until just combined. Stop mixing as soon as no dry flour streaks remain to keep the cake tender.
07 - Pour the batter into the prepared pan and smooth the top with a spatula. Bake for 35–40 minutes, or until a toothpick inserted into the center comes out clean and the top springs back lightly when touched.
08 - Allow the cake to cool completely in the pan set on a wire rack. Do not begin frosting until the cake is fully cooled to room temperature.
09 - Beat the softened cream cheese and butter together in a bowl on medium-high speed until creamy and free of lumps. Gradually add sifted powdered sugar, vanilla extract, and salt. Continue beating on high speed for 2–3 minutes until thick, fluffy, and spreadable.
10 - Spread the frosting generously and evenly over the cooled cake using an offset spatula. For cleaner slices, refrigerate for 20 minutes before cutting into squares.