Preheat the oven to 350°F (175°C). Grease and flour a 9x13-inch baking dish or line it with parchment paper for easier removal.

In a large mixing bowl, beat the granulated sugar, sour cream, and softened butter together until creamy and slightly fluffy.

Add the eggs, mashed ripe bananas, and vanilla extract. Beat until the mixture is smooth and evenly combined.

In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.

Gradually add the dry ingredients to the banana mixture. Mix just until no obvious streaks of flour remain. Avoid overmixing.

Fold in the chopped walnuts, if using.

Transfer the batter to the prepared baking dish and spread it into an even layer.

Bake the Banana Bread Brownies for 25–30 minutes, or until a toothpick inserted near the center comes out clean or with a few moist crumbs.

Place the pan on a wire rack and allow the brownies to cool completely before frosting.

To prepare the frosting, melt the butter in a small light-colored saucepan over medium heat. Continue cooking for about 4–5 minutes, swirling or stirring occasionally, until the butter becomes golden brown and develops a nutty aroma. Watch carefully to prevent burning.

Remove the browned butter from the heat and allow it to cool slightly.

Whisk in the powdered sugar, vanilla extract, and milk until smooth. Add a little additional milk if necessary to achieve an easily spreadable consistency.

Spread the brown butter frosting evenly over the completely cooled brownies.

Allow the frosting to set slightly, then slice into squares or bars and serve.