Preheat your oven to 375°F (190°C). Line a 9x13-inch baking pan with parchment paper.

In a large bowl, cream together granulated sugar, softened butter, and sour cream until light and fluffy. Fold in mashed bananas and vanilla extract.

In another bowl, whisk together all-purpose flour, baking soda, and salt. Fold dry ingredients into the wet banana mixture.

Fold in walnuts if using, pour the batter into the prepared baking pan and smooth the top.

Bake for 30-35 minutes or until golden brown and a toothpick comes out clean.

Let the brownies cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.

To make the frosting, melt the butter in a saucepan until browned, then whisk in powdered sugar, vanilla extract, and milk until smooth.

Pour the frosting over the cooled brownies and spread evenly. Slice into squares and serve.