Preheat the oven to 375℉. Spray a 9x13-inch pan with nonstick spray.
In a large bowl, combine the sugar and butter. Use an electric mixer to beat them together until creamy. Mix in the yogurt (or sour cream), eggs, bananas and vanilla.
Add the flour, baking soda and salt; mix just until moistened.
Spread the batter evenly into the prepared pan and bake for 23 to 28 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs. Meanwhile, make the frosting.
PREPARE THE FROSTING:
In a large skillet over medium heat, heat the butter. Allow it to come to a simmer and cook just until it starts to turn a delicate brown and has a nutty aroma. (Watch it carefully, so you don't burn the butter). Remove the pan from heat immediately.
Whisk in the powdered sugar, vanilla and 2 tablespoons 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. Allow them to rest for at least 10 minutes before slicing and serving.