Preheat oven to 375°F (180°C). Grease a 9x13-inch (25x40 cm) baking pan. For the bars, in a large bowl, beat together sugar, sour cream, butter and eggs until creamy. Beat in mashed bananas and vanilla extract. Add flour, baking soda and salt and mix just until flour is fully incorporated (for about 1 minute).

Spread batter into prepared pan. Bake for 20-25 minutes until golden brown and a toothpick inserted near the center comes out clean. Place the pan on a wire rack to cool.

For the frosting, heat butter in a large saucepan over medium heat until boiling. Continue to heat the butter, swirling occasionally, until the sediments (solids at the bottom of the pan) turn golden brown. Remove from heat and whisk in powdered sugar, vanilla extract or vanilla sugar and milk. Immediately spread the brown butter frosting over the warm bars. Cool completely and cut into bars. Store tightly covered at room temperature for up to three days.