Step 1: Preheat your oven to 375°F (190°C). Prepare a 13 x 9 inch baking pan by greasing it thoroughly. This will prevent the brownies from sticking and ensure easy removal after baking. You can also line the pan with parchment paper, leaving an overhang on the sides for easy lifting.

Step 2: In a large mixing bowl, combine the sugar, sour cream, softened butter, and eggs. Beat these ingredients together until the mixture is light and creamy. This step is crucial for creating a tender brownie base. The sour cream adds moisture and a slight tang that complements the sweetness of the bananas and sugar.

Step 3: Add the mashed bananas and vanilla extract to the mixture. Gently blend until well combined. Ensure the bananas are fully mashed to avoid any lumps in the batter. The vanilla extract enhances the overall flavor profile, adding a warm, aromatic note.

Step 4: In a separate bowl, whisk together the all-purpose flour, baking soda, and salt. This step ensures that the dry ingredients are evenly distributed throughout the batter, preventing clumps and ensuring a consistent texture. Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Be careful not to overmix, as this can lead to tough brownies.

Step 5: Gently fold in the chopped walnuts. Distribute them evenly throughout the batter. The walnuts add a delightful crunch and nutty flavor that complements the banana and brown butter flavors.

Step 6: Spread the batter evenly into the prepared baking pan. Ensure the batter is distributed uniformly for even baking.

Step 7: Bake in the preheated oven for 25 minutes, or until a wooden skewer inserted into the center comes out with moist crumbs attached. The baking time may vary depending on your oven, so it's important to keep an eye on the brownies.

Step 8: Remove the brownies from the oven and allow them to cool slightly until warm (but not hot) before frosting. This will prevent the frosting from melting and running off the brownies.

Step 9: While the brownies are cooling, prepare the brown butter frosting. In a large saucepan, melt the butter over medium heat. Continue cooking until the butter turns a medium brown color and emits a nutty aroma. Watch carefully, as browned butter can burn quickly. The key to perfect brown butter is patience and vigilance. The transformation from melted butter to browned butter is a delicate process that requires constant attention.

Step 10: Remove the saucepan from the heat and whisk in the powdered sugar, adding milk gradually to achieve a smooth, spreadable consistency. The amount of milk required may vary depending on the humidity and the consistency of the powdered sugar. Stir in the vanilla extract.

Step 11: Pour the brown butter frosting over the warm banana bread brownies. Spread the frosting evenly with a spatula, ensuring every corner is covered.

Step 12: Allow the frosted brownies to cool completely before cutting them into squares. This will allow the frosting to set and prevent the brownies from crumbling when cut.