Preheat Oven: Preheat your oven to 350°F (175°C). Grease a 10x15-inch jelly roll pan or a 9x13-inch baking dish if you prefer thicker bars.

Mix Wet Ingredients: In a large mixing bowl, combine the sugar, sour cream, and softened butter. Beat until smooth. Stir in the eggs, mashed bananas, and vanilla extract, blending everything together until fully incorporated.

Combine Dry Ingredients: In a separate bowl, whisk together the flour, baking soda, and salt. Slowly incorporate the dry ingredients into the banana mixture, stirring until everything is just blended.

Add Walnuts (Optional): Fold in the chopped walnuts if desired.

Bake the Bars: Pour the batter into the prepared baking pan, spreading it evenly. Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean. Once done, take the pan out of the oven and let the bars cool completely in the pan.

Prepare the Browned Butter Frosting: In a small saucepan, melt the butter over medium heat. Keep cooking the butter until it turns golden brown and releases a nutty fragrance, which should take about 3-5 minutes. Be sure to monitor it closely to prevent burning. Once done, remove from heat and allow the browned butter to cool slightly. In a mixing bowl, whisk together the powdered sugar, vanilla, and browned butter. Add milk or half-and-half, one tablespoon at a time, until the frosting reaches a smooth, spreadable consistency.

Frost the Bars: Spread the browned butter frosting over the cooled banana bars. Allow the frosting to set slightly before cutting into squares.