01 - Preheat the oven to 350°F. Grease a 9x9-inch baking pan and line it with parchment paper, leaving overhang on two sides for easy removal.
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 large mixing bowl and let cool for 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 vigorously until the mixture is smooth and creamy, about 2 minutes.
05 - Add the eggs one at a time, beating well after each addition until fully incorporated and the batter looks glossy.
06 - Fold in the mashed bananas, sour cream, and vanilla extract, stirring until the mixture is smooth and no large banana chunks remain.
07 - Gradually add the flour mixture to the wet batter, folding gently with a spatula until just combined. Stop mixing as soon as no dry streaks remain to keep the cake tender.
08 - Pour the batter into the prepared pan and use a spatula to spread it evenly, smoothing the top.
09 - Bake for 35 to 40 minutes, rotating the pan halfway through, until the top is golden and a toothpick inserted into the center comes out clean.
10 - Let the cake cool in the pan on a wire rack for at least 1 hour until it reaches room temperature before frosting.
11 - Beat the softened cream cheese and butter together with an electric mixer on medium speed until light and fluffy, about 2 minutes. Gradually add the sifted powdered sugar, vanilla extract, and a pinch of salt, beating on low speed until combined, then increasing to medium-high until the frosting is thick and creamy.
12 - Spread the cream cheese frosting generously over the cooled cake using an offset spatula. Slice into 12 squares and serve.