Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 15x10-inch jelly roll pan.
Beat white sugar, sour cream, 1/2 cup softened butter, and eggs in a bowl with an electric mixer until creamy. Beat in bananas and 2 teaspoons vanilla extract. Add flour, baking soda, and salt; mix for 1 minute. Stir walnuts into batter until evenly incorporated. Spread batter in prepared pan.
Bake in the preheated oven until golden, about 27 minutes. Cool.
Melt 1/2 cup butter in a large saucepan over medium heat until bubbling and golden brown, 3 to 5 minutes. Remove from heat.
Whisk confectioners' sugar, milk, and vanilla extract into melted butter until a smooth frosting consistency is reached. Spread frosting over cooled banana bars.