Preheat the oven to 350 °F (175 °C). Prepare a 10×15-inch (25.5 x 38 cm) rimmed sheet pan by greasing it and lining the bottom with parchment paper. Set it aside for later.
In a medium bowl, whisk together the flour, cocoa, baking soda, baking powder, and salt. Set it aside.
In the bowl of an electric mixer fitted with the paddle attachment, cream the butter and the sugars until they are well combined on medium-low.
Add the eggs, one at a time, beating well between each addition, and scraping down the sides of the bowl as needed.
Add the vanilla and beat again.
With the mixer on low, add the flour mixture alternately with the buttermilk and the grated beets. Continue mixing and scraping down the bowl until the batter is well mixed.
Spread the batter in the prepared pan and bake it for about 40 minutes or until a cake tester poked through the center of the cake comes out clean. The cake will begin to pull away from the sides of the pan when it is done.
Let the cake cool completely before proceeding.
To make the cream cheese frosting
In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and cream cheese until it is light and fluffy.
With the mixer on low, add the other ingredients, except the milk. Increase the mixer speed to medium-high and continue beating. Thin the frosting out with 1–2 tablespoon of milk, as needed, to achieve the desired consistency.
To assemble
Line a baking sheet with parchment paper. With a 3-inch (7.5-cm) cake ring or round cookie butter, cut out 16 rounds of cake. Note one will be made of cake scraps.
For each mini cake, place one layer of cake on the baking sheet. Top with a tablespoon or so of frosting, spread evenly, then another round of cake (pressed down gently) and then another tablespoon of frosting. Repeat with the other rounds of cake to make 8 miniature layer cakes.