01 - Preheat the oven to 350°F. Grease a 9x13 inch baking pan and line it with parchment paper, leaving an overhang on the sides for easy removal.
02 - Place the unsalted butter in a medium saucepan over medium heat. Stir continuously as it melts, foams, and eventually develops a golden amber color with a fragrant nutty aroma, approximately 5–7 minutes. Immediately 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. Set aside.
04 - In a large mixing bowl, combine the browned butter with both granulated and brown sugars. Beat until well incorporated. Add the eggs one at a time, mixing thoroughly after each addition. Stir in the vanilla extract, mashed bananas, sour cream, and whole milk until the mixture is smooth and homogeneous.
05 - Gradually add the dry ingredient mixture to the wet ingredients, folding gently with a spatula until just combined. Be careful not to overmix, as this can produce a dense, tough crumb.
06 - Pour the batter into the prepared baking pan, spreading it into an even layer. Bake for 35–40 minutes, or until the top is golden and a toothpick inserted into the center emerges clean.
07 - Transfer the pan to a wire rack and allow the cake to cool completely in the pan before applying frosting. This step is critical—frosting a warm cake will cause it to melt and slide off.
08 - Using a hand mixer or stand mixer, beat the softened cream cheese and butter together on medium speed until silky smooth and free of lumps. Gradually add the sifted powdered sugar, vanilla extract, and a pinch of salt. Increase the speed to medium-high and continue beating until the frosting is thick, fluffy, and holds its shape.
09 - Spread the frosting generously and evenly over the completely cooled cake using an offset spatula. For cleaner slices, refrigerate the cake for 20–30 minutes before cutting into squares.