Preheat oven to 350°F (175°C) and grease three 8-inch round cake pans.

In a bowl, whisk together the sifted cake flour, baking soda, and salt. Give it a good whisk then set aside.

In a stand mixer fitted with the paddle attachment, cream the butter, oil, and sugar on medium-high speed for 4–5 minutes, until very pale and fluffy. Scrape down the bowl as needed.

Lightly whisk the eggs and egg white together. With the mixer on medium speed, slowly stream the eggs into the batter in intervals, allowing each addition to fully incorporate before adding more and scraping down the sides of the bowl. Mix in the vanilla.

Add the dry ingredients in three additions, alternating with the buttermilk, beginning and ending with the dry ingredients. Mix just until combined. Finish folding by hand to ensure everything is evenly incorporated, but be careful not to overmix.

Divide the batter evenly among prepared pans, smooth the tops, and tap on the counter a few times to remove air bubbles.

Bake for 22–26 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs. Begin checking at 20 minutes if your oven runs hot.

Cool in the pans for 10 minutes, then turn out onto wire racks to cool completely.

In a heatproof bowl, whisk together the eggs and sugar until smooth. Whisk in the lemon juice, zest, and salt.

Place the bowl over a saucepan of gently simmering water (double boiler), making sure the bottom does not touch the water. Cook, stirring constantly, until thickened and glossy. The mixture should coat the back of a spoon and reach 165–170°F. This will take about 30 minutes.

Remove from heat and whisk in the butter until fully melted and smooth. Let cool for about 30 minutes until it has thickened but is still spreadable. You want it still slightly warm when icing the cake.

Place a small spoonful of icing on your cake plate to anchor the cake. Add the first cake layer and spread a generous layer of lemon cheese icing on top. Repeat with remaining layers.

Pour additional icing over the top and gently encourage it to run down the sides, smoothing as you go. Garnish with thin lemon slices just before serving, if desired.

This cake is best after resting 4–6 hours and even better the next day because the icing soaks into the cake layers. Store it covered in the refrigerator but bring to room temperature before serving for the best texture and flavor. If serving slices straight from the fridge, microwave for a few seconds to take the chill off.