Preheat oven to 350 degrees and ensure rack is in the middle position.
Purée beets in food processor until very smooth.
Combine sugar and eggs in mixer and blend well. Add vanilla and peppermint extract. Pour in oil and mix until well blended.
Sift together cocoa powder, flour, baking soda and salt.
Add half of the dry ingredients to the egg mixture, blend, then add half of the beets and blend again. Repeat and blend until batter is well combined.
Pour half of the batter into each of 2 prepared 8 inch cake pans. Prepare pans by buttering and placing parchment paper in the bottom as instructed in above post.
Bake for 30-35 minutes. When a toothpick inserted into the centre of the cake comes out clean, the cake is done. Cool pans on a rack.
Make icing by combining butter, icing sugar, vanilla and cocoa powder in a large bowl. Slowly beat until combined, then increase speed and slowly add milk in spoonfuls until desired consistency is reached. Icing should be stiff enough to hold in place but soft enough to spread around without breaking the cake.
Place cake plate or dinner plate over top of one cake layer and flip so bottom side is up. Remove parchment paper. Ensure cake layer is sitting in the centre of the plate.
Generously spread icing evenly to the outer edges. Using same plate method, flip cake layer out of pan, remove parchment. Gently turn cake layer over and place top side up on bottom layer. Ice cake top and sides. To ice sides, hold spatula in place and spin plate, adding more icing as necessary.