Preheat oven to 375 degrees F and grease a 9x13 inch pan with non-stick cooking spray.
In a large bowl, cream together the sugar and butter. Beat in the sour cream and eggs, bananas and vanilla extract.
Add the flour, baking soda, salt, and mix just until moistened. Stir in walnuts, if using.
Spread batter evenly into prepared pan and bake for 23-28 minutes or until a toothpick inserted in the center comes out clean, or with few crumbs. Make frosting while the bars bake.
For the Brown Butter Frosting:
Heat the butter in a large skillet over medium heat. Allow it to come to a simmer until it starts to turn a delicate brown and has a nutty aroma (watch carefully so you don't burn the butter!) Remove from heat immediately.
Whisk in the powdered sugar, vanilla extract and 1 tablespoon of milk until smooth. (Add a little more powdered sugar, or milk, if needed to thicken or thin the frosting). Pour the warm frosting over the warm bars. Spread gently with the back of a spoon.
Add nuts: add chopped walnuts or pecans to the batter before baking.
Add chocolate chips: add mini milk or semi sweet chocolate chips to batter before baking.
Change the frosting: I love the brown butter frosting but this cream cheese frosting would also be delicious.
Healthier banana bread bars: cut the sugar in half for a lighter version and use 3-4 Tablespoons of butter for the frosting instead of 6.
Make ahead Instructions: bake and frost one to two days ahead of time. Store covered, in the refrigerator.Freezing Instructions: allow them to cool completely and freeze them frosted or unfrosted, right in the pan. Cover well with plastic wrap and aluminum foil before freezing and freeze for up to 3 months. Allow to thaw completely before serving.