Preheat your oven to 350°F (175°C) and prepare a 10x15 jelly roll pan by greasing with butter or cooking spray and lightly flouring.

In a large mixing bowl, combine the sugar, sour cream, softened butter, and eggs. Blend until creamy, about 2-3 minutes.

Stir in the mashed bananas and vanilla extract until just combined, being careful not to over-mix.

Sift together the flour, baking soda, and salt in a separate bowl, then gradually add to the wet ingredients, folding gently until no dry flour patches remain.

Pour the batter into the prepared pan and bake for 18-22 minutes. Check for doneness with a toothpick.

To make the frosting, melt the butter over medium heat until browned, then whisk in powdered sugar, vanilla, and half-and-half until smooth.

Once the bars have cooled slightly, frost them generously with the browned butter frosting and let cool completely.