Preheat your oven to 375°F. Grease and flour a 15×10-inch jelly roll pan.
In a large bowl, beat together the sugar, sour cream, softened butter, and eggs until creamy and well combined.
Blend in the mashed bananas and vanilla extract until incorporated.
Add the flour, baking soda, and salt to the mixture. Blend for about one minute until well combined.
Gently fold in the chopped walnuts.
Spread the batter evenly into the prepared pan, ensuring it reaches all corners.
Bake for 20-25 minutes or until the top turns golden brown and a toothpick inserted in the center comes out clean.
While the brownies are baking, prepare the brown butter frosting. Heat the butter in a large saucepan over medium heat until it begins to boil.
Continue cooking, watching carefully as the butter turns a delicate brown color. Remove from heat immediately once it reaches a nutty, amber hue.
Quickly whisk in the powdered sugar, vanilla extract, and milk until smooth. The consistency should be thicker than a glaze but thinner than traditional frosting.
Allow the brownies to cool slightly, then spread the brown butter frosting evenly over the top.
Let cool completely before cutting into squares and serving.
