Preheat oven to 350 degrees. Grease 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. Spread batter in prepared pan.
Bake in the preheated oven until golden, about 27 minutes.
While bars bake, prepare frosting. Melt 1/2 cup butter in a large saucepan over medium heat, until bubbling and golden brown. Stir with a rubber spatula, scraping from the bottom, until butter is amber colored. Remove from heat. Whisk in powdered sugar, milk, and vanilla until smooth.
Spread over warm bars. Allow bars to cool before slicing and serving.