01 - Preheat oven to 350°F. Grease a 9x13 inch baking pan and line with parchment paper, leaving 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 let cool for 10 minutes.
03 - In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until evenly distributed.
04 - Pour the browned butter into a large mixing bowl. Add the granulated sugar and packed light brown sugar. Beat with a hand mixer or stand mixer on medium speed until well blended and slightly thickened, about 2 minutes.
05 - Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract, mashed bananas, sour cream, and whole milk. Mix until smooth and fully incorporated.
06 - Gradually add the dry ingredient mixture to the wet ingredients. Fold gently with a spatula or mix on low speed until just combined. Stop mixing as soon as no dry streaks remain to keep the cake tender.
07 - Pour the batter into the prepared 9x13 inch pan, spreading it into an even layer. Bake for 35 to 40 minutes, or until the top is golden and a toothpick inserted into the center comes out clean.
08 - Transfer the pan to a wire rack and let the cake cool completely in the pan. Do not begin frosting until the cake is fully cooled to room temperature.
09 - In a large bowl, beat the softened cream cheese and softened butter together using a hand mixer or stand mixer on medium-high speed until creamy and lump-free, about 2 minutes. Gradually add the sifted powdered sugar, vanilla extract, and a pinch of salt. Beat on medium-high speed for 3 to 4 minutes until thick, fluffy, and spreadable.
10 - Spread the frosting generously and evenly over the completely cooled cake using an offset spatula. For cleaner slices, refrigerate the frosted cake for 20 to 30 minutes before cutting into squares.