How to Make It
Preheat oven to 400°F. Place chopped walnuts in a baking dish and toast in the oven until lightly browned, 8 to 10 minutes. Set aside to cool. Reduce oven to 325°F. Grease a glass 15-by-10-inch baking dish 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.
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. Transfer batter to prepared baking dish, using a spatula to level the surface. 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.
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. 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.
Cover Cake evenly with Frosting, and allow Frosting to set before slicing and serving, about 30 minutes.