Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 15x10-inch jelly roll pan.
Make bars: Beat together white sugar, sour cream, softened butter, and eggs in a large bowl with an electric mixer until creamy. Beat in mashed bananas and vanilla. Add flour, baking soda, and salt; mix for 1 minute. Stir in walnuts until evenly incorporated. Spread in the prepared pan.
Bake in the preheated oven until golden and a toothpick inserted into the center comes out clean, about 27 minutes. Place on a wire rack to cool lcompletely.
Make frosting: Melt butter in a large saucepan over medium heat until bubbling and golden brown, 3 to 5 minutes. Remove from heat. Whisk in confectioners' sugar, milk, and vanilla until smooth.
Spread frosting over cooled banana cake. Cut into bars.