Preheat the oven to 350. Line a standard-sized muffin pan with paper liners. If you only have one 12-cup muffin pan, you'll have to bake the cupcakes in batches.
In a large bowl, sift together the flour, sugar, baking powder and salt.
In a liquid measuring cup, measure and whisk together the eggs, egg whites, buttermilk, melted butter, oil and vanilla extract.
Add the liquid ingredients to the dry, and whisk until well combined, about 15 seconds.
Use a 1/4 cup measuring cup to spoon the cake batter into the paper liners, filling them no more than 2/3 full.
Bake the cupcakes on the center oven rack for about 14-15 minutes, until the tops spring back when lightly touched.
Cool the cupcakes in the pan for 2 minutes, then use a spoon to gently lift them out of the pan and onto a cooling rack.
If you have more batter to bake, refill the pan (without stirring up the batter again), and bake the remainder of the cupcakes.
Cool the cupcakes completely before filling and frosting.