Preheat the Oven: Preheat your oven to 375°F (190°C). Grease a 9x13 inch baking pan thoroughly with non-stick cooking spray to ensure the brownies release easily after baking.
Prepare the Batter: In a large mixing bowl, combine the granulated sugar, sour cream, softened butter, and eggs. Utilize an electric mixer to blend the ingredients until they are thoroughly combined and the mixture is smooth. Following this, incorporate the mashed bananas and vanilla extract, mixing until the ingredients are fully integrated into the batter.
Incorporate Dry Ingredients: Gradually add the all-purpose flour, baking soda, and salt to the wet mixture, mixing until the dry ingredients are fully combined and no streaks of flour remain. Fold in the chopped walnuts, ensuring they are evenly distributed throughout the batter.
Bake the Brownies: Pour the prepared batter into the greased baking pan, spreading it evenly. Place the pan in the preheated oven and bake for approximately 20-30 minutes, or until a toothpick inserted into the center of the brownies comes out clean. Upon completion of baking, remove the pan from the oven and allow the brownies to cool slightly in the pan.
Prepare the Frosting: While the brownies are cooling, begin preparing the frosting. In a medium-sized saucepan, melt the butter over medium heat. Continue cooking the butter until it reaches a medium brown color, which will impart a rich, nutty flavor to the frosting. Remove the saucepan from heat and gradually stir in the powdered sugar, adding it one cup at a time and mixing thoroughly after each addition. Finally, incorporate the milk and vanilla extract, stirring until the frosting is smooth and well-blended.
Frost the Brownies: Once the brownies have cooled slightly, pour the frosting over the top, spreading it evenly across the surface. Allow the brownies to cool completely in the pan, ensuring the frosting sets properly. Once cooled, cut the brownies into squares and serve.