Heat oven to 375F. Lightly grease a small oven safe baking dish or pan with nonstick spray.
In a large bowl, mash the banana and mix in the brown sugar, vanilla, and sour cream until creamy. Add flour, baking soda, salt, and mix well. Stir in walnuts.
Spread batter evenly into pan. Bake 20 to 25 minutes or until golden brown. A toothpick inserted into the center should come out clean.
Meanwhile, for frosting, heat butter in a large saucepan over medium heat. 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).