01 - Preheat oven to 350°F (175°C). Grease a 9x13 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 foams and begins to develop golden brown flecks and a nutty aroma, approximately 5–7 minutes. Remove from heat and allow to cool slightly.
03 - In a medium bowl, whisk together all-purpose flour, baking powder, baking soda, and salt until evenly distributed.
04 - In a large mixing bowl, combine the browned butter, granulated sugar, and light brown sugar. Beat until smooth and well incorporated.
05 - Add eggs one at a time, mixing thoroughly after each addition. Stir in pure vanilla extract, mashed bananas, sour cream, and whole milk until the mixture is uniform.
06 - Gently fold the dry ingredient mixture into the wet ingredients. Mix until just combined—avoid overmixing to keep the cake tender.
07 - Pour the batter evenly into the prepared baking pan. Bake for 35–40 minutes, or until a toothpick inserted into the center emerges clean.
08 - Transfer the pan to a wire rack and let the cake cool completely before frosting.
09 - Beat softened cream cheese and softened unsalted butter together until creamy and lump-free. Gradually add sifted powdered sugar, pure vanilla extract, and a pinch of salt. Continue beating until the frosting is thick and fluffy.
10 - Spread the frosting generously and evenly over the cooled cake. For cleaner slices, chill briefly before cutting into squares.