Preheat oven to 350°F. Grease three 6-inch cake pans and line bottoms with parchment paper. Set aside.
In the bowl of a stand mixer fitted with whisk attachment, beat eggs and sugar until pale, about 5-7 minutes. Beat until mixture can form thick ribbons.* (Alternatively, use an electric hand mixer.)
In a large bowl, whisk together flour, baking soda, baking powder, and salt. Set aside. In another bowl, mix together olive oil, lemon zest, lemon juice, buttermilk, and vanilla. Set aside.
With the mixer running on slow speed, alternate adding the dry and wet ingredients in three additions. Start and end with the dry. Scrape down sides of bowl as needed. Mix until just combined. Be careful not to overmix.
Divide batter evenly among prepared cake pans. Bake for 20-25 minutes until toothpick inserted in center of cake comes out clean. Cool cakes in pan for 10 minutes. Run mini offset spatula along edges of cake. Invert and unmold cakes. Cool cakes to room temperature on wire racks.