Place a rack in the center and upper third of the oven. Preheat oven to 375 degrees F.
Thoroughly wash beets under running water, and trim their leaves, leaving about 1/2 inch of stem. Place clean beets in a piece of foil. Drizzle with just a bit of vegetable oil. Seal up foil. Place on a baking sheet in the oven. Roast until beets are tender when pierced with a knife, about 1 hour.
Remove the beets from the oven. Open the foil and allow beets to cool completely. Beets will be easy to peel (just using a paring knife) once completely cooled.
Using a box grater, grate the peeled beets on the finest grating plane. Measure 3/4 cup of grated beets for the cake and 2 tablespoons for the frosting. Set aside.
Reduce the oven temperature to 350 degrees F. Use butter to grease two 8 or 9-inch round baking pans. Trace a piece of parchment paper so it is the same size as the bottom of the cake pan. Cut it out and place inside the cake pan. Butter the parchment paper. Add a dusting of flour to coat the pan. Set pans aside while you prepare the cake.
In the bowl of an electric stand mixer, fitted with a paddle attachment, cream together butter and sugars. Beat on medium speed until pale and fluffy, about 3 to 5 minutes. Beat in eggs, one at a time, for one minute after each addition. Stop the mixer and scrape down the sides of the bowl as necessary. Once eggs are incorporated, beat in beets and vanilla extract until thoroughly combined.
In a medium bowl, whisk together flour, cocoa powder, baking soda, baking powder, and salt.
Add half of the dry ingredients to the butter and egg mixture. Beating on low speed , slowly add the buttermilk. Once just incorporated, add the other half of the dry ingredients. Beat on medium speed until milk and dry ingredients are just incorporated. Try not to overmix the batter. Bowl can be removed from the mixer and mixture folded with a spatula to finish incorporating ingredients. Cake batter will be on the thick side… not pourable.
Divide the batter between the two prepared cake pans. Bake for 23 to 25 minutes (for a 9-inch pan) or 30-32 minutes (for an 8-inch pan). Cake is done when a skewer inserted in the center comes out clean. Remove cakes from the oven and allow to rest for 10 minutes. Invert cakes onto a cooling rack to cool completely before frosting and assembling the cake.
To make the Frosting:
In the bowl of an electric stand mixer, fitted with a paddle attachment, beat cream cheese for 30 seconds, until pliable and smooth. Add the butter and beat for another 30 seconds, until well combined. Stop the mixer and scrape down the bowl as necessary. Beat in the beets. Add the powdered sugar, vanilla extract, milk, lemon juice, and salt. Beat on medium speed until smooth and silky. Refrigerate the frosting for 30 minutes before frosting the cooled cakes.
To assemble the cake, place one layer of cake on a cake stand or cake plate. Top with a generous amount of pink frosting. Spread evenly. Place the other cake on top of the frosting. Top with frosting. Work frosting onto the sides of the cake. You will have extra frosting left over. Refrigerate for an hour before serving (it will make the cake easier to slice). Cake will last, well wrapped in the refrigerator, for up to 4 days.
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