1. Preheat the oven to 400 F and place the quartered beets into a small baking dish, along with about 1/4 cup water. Cover the dish, and bake for about 30-35 minutes, or until the beets are fork-tender. Allow the beets to cool for 5 minutes, then place them in a high-powdered blender, along with a tablespoon or two of liquid and puree until smooth. Measure out 3/4 cup puree to cool. Reserve the remaining liquid from the cooked beets.
2. Turn the heat down to 350 F and spray two 9″ round cake pans with non-stick spray and dust the pans with cocoa powder. In a large mixing bowl, whisk together the almond milk and apple cider vinegar, and allow it to sit for a couple of minutes. Add in the avocado oil and applesauce, and whisk to combine. Add the 3/4 cup beet puree, sugar and vanilla extract. To the boiling water, add the espresso powder, and stir until completely dissolved (set aside).
3. In another large mixing bowl, sift together the flour, cocoa powder, baking soda, baking powder and salt. Add half the dry ingredients to the wet ingredients, stir a few times with a large wooden spoon, then add the espresso. Stir a few more times, then add the remaining dry ingredients. Stir with the spoon until just incorporated. Spread the batter evenly between the two cake pans (use a kitchen scale, if desired).
4. Bake the cakes for 30-35 minutes, checking at 30 minutes with a toothpick to ensure the cake is fully cooked. If the toothpick has batter remaining, bake for the additional 5 minutes. Remove the cakes from the oven, and allow them to cool for 5 minutes, before removing them to a wire rack to cool completely
5. Prepare the vegan buttercream. In a stand mixer fitted with the paddle attachment, or using a handheld electric mixer, whip the softened vegan butter until light and fluffy. Add half of the powdered sugar, followed by 1 tablespoon of reserved beet cooking liquid and mix together. Add the remaining powdered sugar, along with the vanilla extract and mix on high. If the buttercream seems too stiff, add additional beet liquid; if it is too soft, add an additional cup of powdered sugar. Frost the cake!

Frosted cake can remain at room temperature, in an airtight container, for 2-3 days, or 3-4 days in the fridge.

If making ahead of time, the cake can be kept (before frosting) in the freezer for up to 2 months. Thaw overnight before frosting.