To make the cake, preheat the oven to 350°. Spray a 13 x 9-inch baking pan with a flour and oil baking spray.
Whisk together the eggs and yolks, buttermilk, and vanilla. Set aside.
Put the flour, sugar, baking powder, baking soda, and salt into the bowl of a stand mixer. Whisk the mixture together. Then insert the paddle attachment.
Add butter and oil and mix at low speed to combine, then increase the speed and mix until light and fluffy. Reduce speed to low and slowly add the egg mixture.
When the eggs are incorporated, scrape down the bottom of the mixer bowl. Beat on medium-high speed until the batter lightens in color and thickens about 3 minutes. Scrape the batter into the prepared pan and smooth the top with an offset spatula. Rap the pan on the counter 2-3 times to bring up any big air bubbles trapped in the batter.
Bake for 28 to 32 minutes or until a toothpick inserted in the middle comes out with only a few crumbs. Cool completely on a rack.