01 - Preheat oven to 350°F. Grease a 9x9-inch baking pan and line with parchment paper, leaving overhang on two sides for easy removal.
02 - Melt butter in a medium saucepan over medium heat, stirring frequently. Continue cooking until the butter foams, turns a deep golden brown, and releases a nutty aroma, about 5 to 7 minutes. Immediately transfer to a heatproof bowl and let cool slightly.
03 - In a medium bowl, whisk together the all-purpose flour, baking soda, ground cinnamon, and salt until evenly distributed.
04 - In a large mixing bowl, beat the browned butter and granulated sugar together until creamy and well incorporated.
05 - Add eggs one at a time, mixing thoroughly after each addition until the mixture is smooth and emulsified.
06 - Fold in the mashed bananas, sour cream, and vanilla extract, mixing until the mixture is smooth and homogeneous.
07 - Gradually add the dry ingredient mixture to the wet batter, folding gently until just combined. Be careful not to overmix to keep the cake tender.
08 - Pour the batter into the prepared pan and smooth the top with a spatula. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
09 - Allow the cake to cool completely in the pan set on a wire rack before frosting.
10 - Beat the softened cream cheese and butter together in a bowl until light and fluffy. Gradually add the sifted powdered sugar, vanilla extract, and a pinch of salt, beating until the frosting is thick, creamy, and spreadable.
11 - Spread the cream cheese frosting generously over the completely cooled cake. Slice into squares and serve.