Position a rack in the center of the oven if making cakes; if making cupcakes, position racks in the top and bottom thirds of the oven. Preheat the oven to 350°F. If making cakes, spray or butter two round cake pans, each 8 inches in diameter and 2-inches deep, and line the bottoms with parchment; if making cupcakes, line two standard muffin pans with paper or foil liners.
In a liquid measuring cup, mix the oil, water, and apple juice and set aside. In a medium bowl, whisk together the flour, baking powder, and salt and set that aside as well.
In the bowl of a stand mixer with the whisk attachment, combine the sugar, eggs, and vanilla. Whip on medium-high speed until the mixture is thick, has tripled in volume, and makes a fat ribbon that holds its shape for a few seconds on top of the batter when you lift the whisk, about 4 minutes.
With the motor running on medium speed, add the liquid in a slow, steady stream. (The trick here is to incorporate and emulsify the liquid into the eggs without causing the eggs to deflate. Adding the liquid slowly in a constant stream is the name of the game.)
Add the dry ingredients to the egg mixture and mix on low speed just until the batter is smooth and lump-free, about 30 seconds. (Be careful not to overbeat, which will make the cake tough.)
Divide the batter between the cake pans or among the muffin cups. Bake until the cakes are golden, spring back to a light touch, and a toothpick inserted into the center comes out clean, about 35 minutes for cake and about 20 minutes for cupcakes.
Let the cakes cool in the pans for 40 minutes, then invert onto a wire rack. Remove the parchment and let cool completely. Let the cupcakes cool in the pans for 20 minutes before transferring onto a wire rack, then let cool completely.
Reprinted with permission from Sweet Cream & Sugar Cones by Kris Hoogerhyde, Anne Walker, and Dabney Gough, copyright © 2012. Published by Ten Speed Press, a division of Random House, Inc.