Gather ingredients:

Toast walnuts: Preheat oven to 400°F. Place chopped walnuts on a baking pan and toast in the oven until lightly browned, 8 to 10 minutes. Set aside to cool. Reduce oven temperarure to 325°F. Grease a 15-by-10-inch baking pan with cooking spray; set aside.

Make the Cake: In the bowl of a stand mixer, combine flour, sugar, baking powder, baking soda, and salt. Mix on low until evenly distributed. With the mixer on low, add butter 1 tablespoon at a time, waiting until each piece of butter is incorporated before adding the next. When finished mixing, the butter should be completely incorporated into the flour with no lumps remaining.

Combine wet ingredients: In a bowl, whisk together buttermilk, eggs, egg white, oil, and extracts. Add one-third of the liquid mixture to the dry ingredients and mix on low for 2 minutes. Scrape down sides of the bowl with a spatula, add another third of the liquid ingredients and mix on low for another 2 minutes. Scrape the sides of the bowl again, add remaining liquid ingredients and mix for a final 2 minutes. Fold batter a few times with spatula to ensure that no unincorporated bits are stuck to the bottom and sides of the bowl.

Add batter to pan: Transfer batter to prepared baking pan, using a spatula to level the surface.

Bake cake: Bake until lightly golden brown across the surface and until a toothpick inserted in the center comes out clean, 30 to 35 minutes. Allow cake to partially cool before adding the Frosting.

Make browned butter: While cake is baking, melt butter for Frosting in a saucepan. Allow to foam and boil until butter browns and nutty aromas come from the saucepan, about 5 minutes.

Make Frosting: Transfer browned butter to mixer. Add milk, powdered sugar, and vanilla. Mix on low until sugar is incorporated, scraping down the sides of the bowl as needed. Beat on medium-high for 3 minutes. Add toasted walnuts to Frosting and stir to combine.

Add Frosting to Cake: Cover Cake evenly with Frosting, and allow Frosting to set before slicing and serving, about 30 minutes.