1. Cover beets with 2 inches water in a pot. Bring to a boil. Reduce heat, and simmer until very tender when pierced with the tip of a sharp paring knife, about 30 minutes. Drain. Puree beets in a food processor until smooth.
2. Preheat oven to 350 degrees. Whisk together flour, sugar, cocoa powder, baking soda, and 3/4 teaspoon salt in a large bowl. Whisk in eggs, water, oil, vanilla, and 1 1/4 cups beet puree (reserve remaining puree for another use).
3. Coat a 9-inch round cake pan (3 inches deep) with cooking spray. Line bottom with parchment, and coat with spray. Pour batter into pan. Bake until a toothpick inserted into the center comes out clean, about 45 minutes. Let cool in pan on a wire rack for 20 minutes. Turn out cake from pan, and discard parchment. Let cool completely, right side up.
4. Trim top of cake using a serrated knife to create a level surface. Transfer cake, cut side down, to a platter. Sprinkle powdered sugar over the top and serve.