Grease and line a 13 by 9-inch (33 x 23 cm) baking pan. See note 7
Preheat the oven: Preheat the oven to 350°F/180°C (160°C fan)
Combine eggs and yolks, buttermilk, and vanilla in a bowl or measuring jug and whisk or beat until smooth. Set aside
In a separate large bowl, whisk together the cake flour, sugar, baking powder, baking soda, and salt.
Add butter and oil and mix on low speed until combined, about 30 seconds. The mixture will have a crumbly texture.
With mixer running, slowly add egg mixture.
When mixture is fully incorporated, stop mixing and scrape down bowl with a rubber spatula.
Beat on medium-high speed until the batter is pale, smooth, and thick, about 3 minutes.
Transfer the batter to the greased and lined pan and smooth the top. Tap the pan firmly on counter to release any large air bubbles.
Bake until toothpick inserted in center comes out clean 25 to 30 minutes.
Turn out the cake onto a wire rack and remove the baking paper.
Allow to cool completely before frosting.