Preheat oven to 350 °F (177 °C). Grease and flour three 9-inch round cake pans or 2 quarter sheet pans or one 9x12-inch pan.
To make the cake, beat the egg whites until they're stiff. Transfer them to another bowl, then clean mixing bowl and beater.
5 eggs
Combine the butter, vegetable oil, and granulated sugar in the mixing bowl and mix until light and fluffy. Add the egg yolks and vanilla, then beat until smooth.
1/2 cup unsalted butter, 1 cup vegetable oil, 1 cup granulated sugar, 1 tablespoon vanilla extract
Add the coconut and beat to combine.
1 cup sweetened coconut flakes
Combine the flour, baking soda, and baking powder in a bowl then alternate adding the flour mixture and the buttermilk, mixing for a few seconds after each addition.
Add the egg whites and fold them gently in with the rest of the batter.
Pour the batter onto the greased pans and spread it out. Bake cakes for 15-18 minutes or until a toothpick comes out clean.
Turn the cakes out of the pans and allow them to cool completely.
In the meantime, you can make your icing. To make, combine the cream cheese, butter, vanilla, and powdered sugar in a mixing bowl. Beat until very light and fluffy.