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 - Place unsalted butter in a medium saucepan over medium heat. Stir continuously as the butter melts, foams, and eventually develops a golden brown color with a nutty aroma, approximately 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 thoroughly incorporated. Add eggs one at a time, mixing well 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 until just combined. Be careful not to overmix, as this can result in a dense texture.
06 - Pour the batter evenly into the prepared baking pan. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
07 - Allow the cake to cool completely in the pan set on a wire rack. Do not apply frosting while the cake is still warm.
08 - Using a hand mixer or stand mixer, beat the softened cream cheese and softened butter together until creamy and completely smooth. Gradually add the sifted powdered sugar, vanilla extract, and salt, continuing to beat until the frosting becomes thick and fluffy.
09 - Spread the frosting generously and evenly over the completely cooled cake. For cleaner slices, refrigerate briefly before cutting into squares.