Grease a 9x13-inch baking dish and preheat the oven to 375 F.
Whisk together the sugar, butter, sour cream, and eggs until smooth and creamy.
Fold in the mashed banana and vanilla extract until well combined.
Add the flour, baking soda, salt, and chopped walnuts. Do not overmix.
Pour the batter into the baking dish and bake for 25 mins or until golden brown on top and a toothpick inserted in the middle comes out clean.
While the brownies are baking, make the frosting. Melt the butter in a medium-sized saucepan over medium-low heat and cook until it turns a brown color.
Turn off the heat and mix in the powdered sugar, vanilla extract, and milk until smooth.
Let the brownies cool slightly and spread the frosting evenly on top. Let them cool down completely before cutting and serving.