Preheat your oven to 375°F (190°C) and grease a 15x10 inch pan lightly with butter or cooking spray, then sprinkle a bit of flour to prevent sticking.
In a large mixing bowl, combine 1½ cups of sugar, 1 cup of sour cream, and ½ cup of softened butter. Mix until creamy.
Add 1¾ cups mashed bananas and 2 teaspoons of vanilla extract to the mixture and mix until well combined.
In another bowl, whisk together 2 cups of all-purpose flour, 1 teaspoon of baking soda, and ¾ teaspoon of salt. Gradually incorporate this into the wet ingredients. Stir in ½ cup of chopped walnuts if desired.
Pour the batter into the prepared pan and spread it evenly. Bake for about 20-25 minutes until golden brown on top.
In a saucepan over medium heat, melt ½ cup of butter until browned. Remove from heat and whisk in 4 cups of powdered sugar, 1½ teaspoons of vanilla extract, and 3 tablespoons of milk to create a smooth frosting.
Pour the browned butter frosting over the warm brownies and let them cool completely before slicing into squares.