Grease and line a 13 by 9-inch (33 x 23 cm) baking pan. See note 7

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.

Beat the butter on medium speed with a hand mixer until it is smooth, about 1 minute.

Add the powdered sugar and cocoa powder to the bowl and beat until combined.

Pour in the milk and vanilla extract and continue beating until well combined, scraping down the sides as needed, about 2 minutes.

Leftovers can be stored covered at room temperature for up to 3 days or in the fridge for up to 5 days.

The cake and frosting can be made up to three days ahead, refrigerated, and the cake frosted the day you wish to serve it. For storing, wrap the completely cooled cake in cling wrap and transfer the frosting to an airtight container.

When you're ready to decorate the cake, remove the frosting from the fridge and warm at room temperature until soft enough to spread easily.