Preheat an oven to 375°F and prepare a 9 X 13 pan by spraying with baking spray or lightly greasing.
Combine the butter, white sugar, and brown sugar in a large bowl and mix until well combined.
1/2 cup salted butter, 3/4 cup granulated white sugar, 3/4 cup light brown sugar
Add the eggs, one at a time, mixing well between additions. Scrape the bowl as needed.
2 large eggs
Add the vanilla and sour cream and beat until completely combined.
1 tablespoon vanilla extract, 1 cup sour cream
Add the flour and baking soda and mix just until the flour is incorporated.
2 cups all purpose flour, 1 teaspoon baking soda
Add the bananas and mix until smooth.
1 3/4 cups mashed bananas
Pour the batter into the prepared pan and bake for 25 to 30 minutes or until a toothpick inserted into the center of the brownies comes out clean.
During the last 5 minutes of baking, prepare the frosting.
Frosting
Place the butter into a medium sauce pan and set the heat to medium-high. Heat the butter until it starts to bubble, stirring frequently until the butter has turned a medium to dark blonde color, about 2 to 3 minutes. Remove from the heat.
1/2 cup butter
Add the powdered sugar and mix to combine, add the vanilla and milk and stir until smooth and creamy.
Top the brownies with the frosting while the brownies and frosting are both still warm, this will make spreading out the frosting easier.
Notes
If the frosting is done before the brownies are, move the frosting back to the still-warm burner to keep it soft for longer. It can also be held on a "keep warm" setting if your stove has one.
Work quickly when frosting the brownies, otherwise the frosting will start to form a skin as it cools.