In a saucepan over medium heat, melt the butter. Whisk frequently, scraping the bottom. As the butter foams, continue whisking and scraping to release any brown bits that may stick to the bottom. Once the butter has sufficiently browned (not burned) and smells heavenly, remove from heat.
Preheat oven to 375 degrees F. Line a 15 1/2 x 10 1/2 inch jelly roll pan with parchment paper or grease with butter.
Add browned butter, sugar, sour cream, and eggs to the bowl of a stand mixer. Using the paddle attachment, mix on low speed for about a minute. Scrape sides, add bananas and vanilla, and continue to mix on low for another 30 seconds.
In a separate bowl, combine flour, salt, and baking soda. Add dry ingredients to the wet. Start mixing on low speed just to combine and then flash on high speed for about 10 seconds to fully mix. Scrape sides, add nuts, and mix to combine.
Spread batter in pan and bake in preheated oven until light brown, about 20 to 25 minutes.
Allow bars to fully cool before you frost with brown butter vanilla bean frosting and cut into individual servings.
These store really well in an airtight container in the freezer.