Zest and juice the lemons - 2 medium lemons were enough to give me all the zest and juice needed for the recipe - set the juice and zest aside for later.
Preheat the oven to 350F and prepare 2 6-inch round cake pans by greasing them and lining the bottoms with parchment paper.
To a small bowl, add the flour, baking powder, and baking soda. Whisk to combine and set the bowl aside.
To a 2-cup measuring cup, add the milk, yogurt, vanilla extract, and lemon juice. Whisk to combine and set aside.
To a medium bowl, add the sugar and lemon zest. Using your fingers, rub the zest into the sugar until all of the sugar is coated in zest. Add the butter and oil to the bowl. With a handheld mixer, blend at medium-high speed for about 2 minutes.
Add half of the liquid mixture to the butter mixture and blend at medium speed for 15 seconds. Add half of the flour mixture to the bowl and blend at medium speed for 15 seconds.
Add the rest of the liquid mixture to the bowl and blend at medium speed for 15 seconds, add the rest of the flour mixture and blend at medium speed until no more flour can be seen.
Give the batter a few good stirs with a spatula and then divide the batter evenly between the two pans.
Bake the cakes for 40 minutes or until set – edges are pulling away from the pan and a toothpick inserted comes out clean.
Let the cakes rest in the pans for 10 minutes and then turn them out onto a cooling rack and remove the parchment paper. Make sure the cakes have cooled to room temperature before frosting them.