Preheat oven to 350° F and position a rack in the center of the oven. Butter two 8-inch cake pans and line them with parchment paper.

Peel and roughly chop beets. Add beets to food processor, and begin to process, gradually adding buttermilk, vinegar, lemon juice, and vanilla extract until a frothy, smooth mixture forms.

Combine cake flour, cocoa powder, baking powder, cream of tartar, and salt together in a bowl, whisking or stirring until evenly combined.

In a large bowl, beat the sugar, oil, and butter together until creamy. Add the eggs one at a time, mixing for 30 seconds after each addition.

Alternating, add the flour mixture and beet mixture, beginning and ending with the flour mixture. Divide between the prepared pans and bake until cakes are baked through, 20 to 30 minutes. If a toothpick inserted in center comes out clean, the cake is done.

Cool the cakes on a rack for 20 minutes, then invert onto the rack to cool completely.

Place the cream cheese, honey, and vanilla extract in the bowl of a stand mixer and beat on medium speed.

Add milk one teaspoon at a time until the frosting reaches desired creaminess.

Frost cake immediately and then chill until serving.