01 - Preheat oven to 350°F. Grease a 9x13 inch baking pan and line with parchment paper, leaving an overhang on the sides for easy removal.
02 - Melt butter in a medium saucepan over medium heat. Stir continuously as it begins to foam, watching carefully until it turns a golden amber color and releases a nutty, toasted aroma, about 5 to 7 minutes. Remove from heat immediately 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.
04 - In a large mixing bowl, combine the browned butter with granulated sugar and brown sugar. Beat until 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 smooth.
05 - Gradually add the flour mixture to the wet ingredients, folding gently until just combined. Take care not to overmix, as this can produce a dense, tough crumb.
06 - Pour the batter evenly into the prepared pan, smoothing the top with a spatula. Bake for 35 to 40 minutes, or until a toothpick inserted into the center emerges clean and the top springs back when lightly pressed.
07 - Allow the cake to cool completely in the pan set on a wire rack. Do not begin frosting until the cake has reached room temperature.
08 - Using a hand mixer or stand mixer, beat the softened cream cheese and butter together on medium speed until creamy and free of lumps. Gradually add sifted powdered sugar, vanilla extract, and a pinch of 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 15 to 20 minutes before cutting into squares.