Preheat your oven to 375°F (190°C) and prepare a 9x13-inch baking pan by lining it with parchment paper.

In a large mixing bowl, cream together granulated sugar, softened butter, and sour cream until light and fluffy. Stir in mashed ripe bananas and vanilla extract.

In a separate bowl, whisk together all-purpose flour, baking soda, and salt. Gradually fold the dry mixture into the wet ingredients, adding chopped walnuts if desired.

Pour the batter into the prepared baking pan and bake for 30-35 minutes, until the top is golden brown and a toothpick comes out clean.

Allow the brownies to cool in the pan for about 10 minutes before lifting them out to cool completely on a wire rack.

In a saucepan, melt unsalted butter over medium heat, swirling until golden brown. Remove from heat and whisk in powdered sugar, vanilla extract, and milk until smooth.

Once the brownies are cool, frost with the brown butter frosting and slice into squares for serving.