To make the Lemon Cream, combine cream, sugar, peel, and juice in a large bowl and beat with an electric mixer or large whisk until soft peaks form. Cover and chill up to 4 hours ahead. Whisk before spreading on cake.
Preheat oven to 350° F. Grease three 6 x 2-inch round cake pans. Line bottoms of pans with parchment paper; grease the paper. Set pans aside.
In a large mixing bowl, beat the eggs and sugar with an electric mixer about 5 minutes or until pale and thick ribbons form.
In another large bowl whisk together the flour, baking soda, baking powder, and salt. In a medium bowl combine the olive oil, lemon peel, lemon juice, and buttermilk.
Beat the vanilla into beaten egg mixture on low speed. With mixer on low speed, add the dry and wet ingredients in three additions, starting with dry and ending with wet. After the last addition, turn mixer off and whisk until combined. Divide batter among prepared pans, filling each half full (1 2/3 cups batter each).
Bake for 20 to 25 minutes or until a toothpick inserted near the centers comes out clean and cakes are golden and pull away from sides. Remove and cool in pans on wire racks for 10 minutes. Remove from pans. Cool completely.
To assemble, place one cake layer on a serving plate. Spread top of cake with 3/4 cup of the cream, pushing just past the edges. Place another cake layer on top. Spread with another 3/4 cup cream, pushing just past the edges. Top with remaining cake layer. Spread with the remaining frosting. Use an offset spatula or the flat side of a knife to skim the sides of the cake, pulling off any frosting overhang.