01 - Preheat oven to 350°F (175°C). Grease a 9×13 inch baking pan and line with parchment paper, leaving overhang on the sides for easy removal.
02 - Melt unsalted 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 fragrant nutty aroma, about 5–7 minutes. Remove from heat immediately and let cool slightly.
03 - In a medium bowl, whisk together 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 granulated sugar and packed light 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 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 pan, spreading it into an even layer. Bake on the center rack for 35–40 minutes, or until a toothpick inserted into the center comes out clean with only a few moist crumbs.
07 - Allow the cake to cool completely in the pan set on a wire rack. Do not begin frosting until the cake reaches room temperature, as residual heat will melt the frosting.
08 - Using a hand mixer or stand mixer, beat softened cream cheese and softened 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 holds its shape.
09 - Spread frosting generously and evenly over the completely cooled cake using an offset spatula. For cleaner slices, chill the frosted cake briefly in the refrigerator for 20–30 minutes before cutting into squares.