Preheat the oven to 350 F. Grease and flour an 8 inch round cake pan that is 3 inches high.
Set the sugar in a large bowl and zest the lemons over it. Rub the zest into the sugar with your fingers until it’s all worked in and the sugar is fragrant with the lemon oil.
Crack the eggs into the sugar and use a hand mixer or a stand mixer to beat until light and fluffy, the color will change to very pale. Add the vanilla and salt as it beats.
Once its pale, keep the mixer on and slowly pour in the olive oil. Beat until fully combined.
Over the batter, sift in the cake flour. Add the baking powder and soda. Turn the mixer on and beat - once half of the flour is combined into the batter, keep the mixer on and slowly pour in the buttermilk and then the lemon juice. Stop as soon as all the ingredients are combined.
Pour the batter into the prepared pan. Bake for 30-40 minutes, until a cake tester comes out clean.