Preheat your oven to 375°F (190°C) and grease and flour a 15x10-inch jelly roll pan.

In a large mixing bowl, combine the sugar, sour cream, butter, and eggs using a whisk until creamy and smooth. Add the mashed bananas and vanilla extract, stirring well.

Gently fold in the all-purpose flour, baking soda, and salt. Mix until just combined, being careful not to overmix. If using walnuts, fold them in now.

Pour the brownie batter into the prepared pan, spreading it evenly. Bake for 20-25 minutes until the edges are golden and a toothpick comes out with moist crumbs.

For the frosting, melt the butter in a saucepan over medium heat until golden brown and nutty. Remove from heat and whisk in powdered sugar, vanilla, and milk until smooth.

Let the brownies cool in the pan for 10 minutes before frosting them with the brown butter icing while they're still warm.