Place a rack in the center of the oven. Preheat the oven to 350F.
Grease and flour either two 8 or 9-inch round cake pans or two 12-count muffin tins. You can line the muffin cups with paper liners in place of greasing. Set aside.
Beat together the butter, oil, sugar, baking powder, and salt together with a mixer until well-combined, about 3 minutes on medium-high.
½ cup butter, ⅓ cup oil, 1 ½ cups granulated sugar, 2 ½ teaspoons baking powder, ½ teaspoon salt
Scrape down the sides.
Add the vanilla, eggs and then egg yolks one at a time, and mix until well-combined.
4 large eggs, 2 egg yolks, 1 tablespoon vanilla extract
Scrape down the sides.
Stir in half of the flour, then all of the milk, and then the remaining flour, scraping the bowl after each addition.
3 cups cake flour, 1 cup whole milk
Divide the batter evenly between the prepared pans.
Bake the cake until just golden brown. For cupcakes - 20-30 minutes. For 8-inch and 9-inch round cake pans 35-45 minutes. The cake should have a few moist crumbs attached when a toothpick is inserted in the center but there shouldn't be wet batter.
For round cake pans, cool completely in the pan, then loosen the edges with a knife and gently remove from the pan.
For cupcakes, let cupcakes to cool in the pans for 5-10 minutes, then remove to a wire rack to cool completely.
Notes
Oil: A neutral oil such as canola oil is best.
Flour: Properly measure the flour by aerating, spooning, and then leveling. Check out this tutorial to make a homemade cake flour substitute.
Milk: Anything from skim to whole milk will work. However, whole milk will give you moister, richer cake.