01 - Preheat oven to 350°F. Grease a 9x13 inch baking pan and line it with parchment paper, leaving 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 brown color with a nutty aroma, approximately 5 to 7 minutes. Remove from heat and let 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 granulated sugar and brown sugar. 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 smooth.
05 - Gradually add the dry ingredient mixture to the wet ingredients. Fold gently with a spatula until just combined, being careful not to overmix.
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 before frosting.
08 - Using a hand mixer or stand mixer, beat the softened cream cheese and butter together until creamy and smooth. Gradually add the sifted powdered sugar, vanilla extract, and salt. Continue beating until the frosting is thick and fluffy.
09 - Spread the frosting generously and evenly over the completely cooled cake. For cleaner slices, chill the frosted cake briefly before cutting into squares.