For a 9x13-inch cake, double the recipe (both cake and frosting) and bake for 45-50 minutes. The cake stays moist and flavorful for a few days, so you can make it 1-2 days ahead.
In a medium bowl whisk eggs, granulated sugar, and brown sugar until well combined for about a minute. Add oil and whisk until combined. Add applesauce, buttermilk, and vanilla extract and mix until well combined. Pour the mixture into the flour mixture and mix as little as possible, just until combined and smooth.