Preheat oven to 375 F.

Grease and flour 15×10-inch jelly roll pan. In a large bowl, beat together sugar, sour cream, butter, and eggs until creamy. Blend in bananas and vanilla extract. Sift together flour, baking soda, and salt. Stir dry ingredients into the batter with a wooden spoon until just combined. Pour batter into prepared pan. Bake until golden brown and a knife inserted in the middle comes out clean, about 20-25 minutes.

While the bars are baking, make the frosting. Heat butter in a small saucepan over medium heat until boiling. Leave the butter over the heat until it turns a delicate brown and smells toasted and nutty. Remove from heat. Add powdered sugar, vanilla extract, milk, and cinnamon. Whisk together until smooth (it should be thicker than a glaze but thinner than frosting).

Using a spatula, spread the brown butter frosting over the warm bars (the frosting will be easier to spread while the bars are still warm). When the bars cool complete, refrigerate until ready to serve.