Preheat your oven to 350°F (175°C). Grease a 10x15 jelly roll pan with butter or cooking spray, then lightly flour the surface.

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

Stir in the mashed bananas and pure vanilla extract until just combined.

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

Pour the batter into the prepared pan and spread evenly. Bake for 18-22 minutes, checking for doneness with a toothpick.

For the frosting, melt butter in a saucepan over medium heat, stirring until it turns a rich amber color. Remove from heat and whisk in powdered sugar, vanilla, and half-and-half.

Once bars have cooled slightly, spread the browned butter frosting over the tops. Allow to cool completely on a wire rack.