Preheat oven to 375 degrees F. Grease a 9x13 deep or a 10x15 inch pan (I use a glass dish).
In a saucepan over medium heat, melt the butter. Add walnuts. Whisk frequently, scraping the bottom. As the butter foams, continue whisking and scraping to release any brown bits that may stick to the bottom. As the butter foams, you can slowly reduce the heat to avoid burning. Once the butter has sufficiently browned (not burned) and smells heavenly, remove from heat and allow to cool to the touch. ***Note: Browned butter can go from perfect to burnt quickly so be sure to keep your eye on it.
Add granulated and brown sugars as well as the vanilla to the warm browned butter. Stir to combine.
Add sugar mixture along with sour cream, eggs, and mashed banana to the bowl of a stand mixer. Using the paddle attachment, mix on low speed for about a minute. Scrape sides 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 and briefly mix again to combine.
Spread batter into prepared pan and bake in preheated oven until light brown on top and fully cooked thru when using a cake tester to determine if the batter is still wet. If using a 9x13 pan, batter will be quite full. Bake for about 35 minutes at which time the top should be browned. If, at that point, the cake jiggles when you move the pan or a cake tester comes out wet, loosely set a piece of foil on top to prevent browning and reduce oven temperature to 325°F and continue cooking until set, an additional 5-15 minutes. If you are using a 10x15 inch pan, the cake will be much thinner and will cook much faster, so remove from oven when browned and fully set. While cake is baking, prepare frosting.