Heat oven to 375°F Grease and flour 15×10-inch jelly roll pan. For the bars, in a large bowl, beat together sugar, sour cream, butter, and eggs until creamy. Blend in bananas and vanilla extract. Add flour, baking soda, salt, and blend for 1 minute. Stir in walnuts.
Spread batter evenly into pan. Bake 20 to 25 minutes or until golden brown.
Meanwhile, for frosting, heat butter in a large saucepan over medium heat until boiling. Let the butter turn a delicate brown and remove from heat immediately.
Add powdered sugar, vanilla extract and milk. Whisk together until smooth (it should be thicker than a glaze but thinner than frosting). Using a spatula, spread the brown butter frosting over the warm bars (the frosting will be easier to spread while the bars are still warm).