01 - Preheat the oven to 350°F (175°C). Grease a 9x9-inch cake pan and line it with parchment paper.
02 - Melt the 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 mixing bowl and let cool slightly.
03 - In a medium bowl, whisk together the flour, baking soda, cinnamon, and salt until evenly distributed.
04 - In a large bowl, beat the browned butter and granulated sugar together until creamy and thoroughly combined.
05 - Add the eggs one at a time, mixing well after each addition until fully incorporated.
06 - Fold in the mashed bananas, sour cream, and vanilla extract. Mix until the mixture is smooth and homogeneous.
07 - Gradually fold the dry ingredient mixture into the wet batter, mixing just until combined. Avoid overmixing to keep the crumb tender.
08 - Pour the batter into the prepared cake 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 applying frosting.
10 - Beat the softened cream cheese and butter together until smooth and fluffy. Gradually add the sifted powdered sugar, vanilla extract, and a pinch of salt. Continue beating until the frosting is thick, creamy, and spreadable.
11 - Spread the cream cheese frosting generously and evenly over the cooled cake. Slice into squares and serve.