01 - Preheat the oven to 350°F. Grease a 9x9-inch baking pan and line it with parchment paper, leaving slight overhang on the sides for easy removal.
02 - Place the butter in a medium saucepan over medium heat. Stir frequently as it melts, foams, and eventually turns a golden amber color with a nutty aroma, approximately 5 to 7 minutes. Immediately transfer to a mixing bowl and let cool for about 10 minutes.
03 - In a medium bowl, whisk together the all-purpose flour, baking soda, ground cinnamon, and salt until evenly distributed.
04 - Add the granulated sugar to the browned butter and beat with an electric mixer until the mixture is creamy and well incorporated.
05 - Add the eggs one at a time, beating thoroughly after each addition until fully combined before adding the next.
06 - Stir in the mashed bananas, sour cream, and vanilla extract. Mix until the mixture is smooth and homogeneous.
07 - Gradually add the dry ingredient mixture to the wet ingredients, folding gently with a spatula until just combined. Avoid overmixing 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 - In a mixing bowl, beat the softened cream cheese and butter together with an electric mixer until smooth 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 and evenly over the completely cooled cake. Slice into squares and serve.