Preheat the oven to 375°F (190°C). Grease a 9×13 inch baking pan with non-stick cooking spray.
In a large mixing bowl, cream the butter and granulated sugar together until light and fluffy. Incorporate the Greek yogurt (or sour cream) and eggs, mixing well. Fold in the mashed bananas and vanilla extract.
Gradually add the all-purpose flour, baking soda, and salt, mixing just until the dry ingredients are moistened. If desired, stir in chopped walnuts.
Evenly spread the batter into the prepared pan. Bake for 23-28 minutes or until a toothpick inserted into the center comes out clean or with a few crumbs. While the bars bake, prepare the frosting.
For the brown butter frosting, melt the butter in a skillet over medium heat. Let it simmer until it reaches a delicate brown color and a nutty fragrance. Remove from the heat immediately.
Whisk the powdered sugar, vanilla extract, and 1 tablespoon of milk into the browned butter until smooth. Adjust consistency by adding more powdered sugar or milk if needed. Pour the warm frosting over the warm bars and spread it gently using the back of a spoon.