Adjust oven rack to middle position and heat oven to 350 degrees. Grease and flour 13 by 9-inch baking pan. Combine eggs and yolks, buttermilk, and vanilla in 2-cup liquid measuring cup and whisk with fork until smooth.

Combine flour, sugar, baking powder, baking soda, and salt in bowl of stand mixer fitted with paddle. Mix on low speed until combined, about 20 seconds. Add butter and oil and mix on low speed until combined, about 30 seconds. Increase speed to medium and beat until lightened, about 1 minute. Reduce speed to low and, with mixer running, slowly add egg mixture. When mixture is fully incorporated, stop mixer and scrape down bowl and paddle thoroughly. Beat on medium-high speed until batter is pale, smooth, and thick, about 3 minutes. Transfer batter to prepared pan and smooth top. Tap pan firmly on counter 5 times to release any large air bubbles.

Bake until toothpick inserted in center comes out with few crumbs attached, 28 to 32 minutes. Let cake cool completely in pan on wire rack, about 2 hours.