Preheat the oven to 350°F. Set out a 9 x 13-inch baking dish or two 9-inch round baking pans. Grease the pans on all sides.
Set a small pot over medium heat. Add ½ cup of butter to the pot. Once melted, stir and watch the butter as it starts to turn brown. The moment the butter is a light golden brown and smells toasted and nutty, remove it from heat. Place the small pot in a dish of ice to cool the butter down quickly.
Meanwhile, place 1 cup of chopped pecans in a dry skillet. Set over medium heat. Shake and toast pecans for 2 to 3 minutes, until they start to smell rich and nutty. Then remove from the heat and allow the chopped pecans to cool.
Set out a mixing bowl for the dry ingredients. Combine the cake flour, salt, baking powder, and baking soda. Whisk well to make sure there are no clumps of baking powder or baking soda in the mix.
Once the butter has cooled and started to solidify, set out a large stand mixer. Combine the cooled brown butter, vegetable oil, brown sugar, and white sugar. Beat on a high for 3-4 minutes, to break down the sugar crystals.
Then scrape the bowl with a rubber spatula, and mix in the eggs, additional egg white, and vanilla extract.
With a mixer running on low, alternate mixing in the flour mixture and the buttermilk until a smooth batter has formed. Scrape the bowl with a rubber spatula one last time. Then turn the mixer on low and mix in the cooled toasted pecans.
Scoop the cake batter into the prepared pan(s) and smooth it out evenly. Bake for 22-25 minutes for round cake pans or 35-40 minutes for a sheet pan. Once a toothpick inserted in the center of the pan comes out clean, the cake is ready to take out of the oven. Allow the cake to cool completely before proceeding.
For the frosting: Place the cream cheese, softened butter, brown sugar, vanilla extract, and salt in a (clean) bowl of an electric mixer. Beat on high for 2 to 3 minutes to break down the sugar and make sure there are no clumps of cream cheese or butter left.
Scrape the bowl with a rubber spatula, then turn on low and slowly beat in the powdered sugar. If needed, you can add a little additional powdered sugar to firm up the frosting.
Once the cake is fully cooled, frost the top of the cake. If making a layer cake, scoop ¾ cup of frosting over one cake layer, then spread it out evenly to the edges. Place a second cake layer on top of the first, and scoop the rest of the frosting on top of the cake. Use a spatula to work the frosting over the top and down the sides of the cake.
You can decorate the top of the cake with whole pecan halves, forming a checkerboard pattern as we did, or any other pattern you like! We also piped remaining frosting around the edge of the sheet pan. This can be done around the top of a layer cake or at the base as well.
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Notes
You can make this cake a sheet cake, as seen in the photos, or you can make it in two 9-inch cake pans and stack them to make a layer cake. There's enough frosting for both methods.Cover the cake with a glass dome or cover it loosely with plastic wrap. Store at room temperature for up to 3 days or in the refrigerator for up to a week. If you choose to refrigerate the cake, it's important to remember that all cakes taste best at room temperature. This also increases the quality of the cake texture as well. So if you've chilled the cake, let it sit out for 30-60 minutes before serving.