Preheat oven to 375.
Flour and grease a 9x13 baking dish.
In a large bowl, beat together sugar, sour cream, eggs, and butter until creamy.
Blend in bananas and vanilla.
Add flour, baking soda, and salt. Mix until completely blended.
Spread into pan and bake for 25 to 28 minutes or until golden brown. (Be careful not to overcook. You don't want to dry them out. They will continue to bake as they rest in the pan.)
In the last 10 minutes of baking, make the frosting.
Heat butter in a medium sauce pan over medium heat until boiling.
Watch carefully and stir the butter to prevent it from scorching. Once the butter turns a toasty brown color, remove from heat immediately.
Add powdered sugar, vanilla extract, and milk. Mix until smooth. It should be thicker than a glaze but a little thinner than a frosting.
Spread on Banana Bread while the bread is still warm.
Allow to completely cool before serving. Cut into bars.