Peel and dice one large beet. Place the pieces in a saucepan with water to cover and boil until soft. (I’ve heard that you can buy canned beets, so you may want to skip this step–and the resulting red hands–by using pre-cooked beets.) Allow the beets to cool, and then drain them, reserving the red water for another purpose. Put the drained beets into the food processor with 1/4 cup (clear) water, and process until pureed.
Preheat the oven to 325 degrees. Oil or spray your cooking pan(s).
Put the pureed beets into a 2-cup measure. Add enough apple sauce to reach the 2-cup line. Add the 2 tablespoons water, vanilla extract, and apple cider to the beets and mix well.
Mix the dry ingredients together; then add the beet mixture and stir until well-combined. Bake for 35-60 minutes, depending on the size of pan you use: more for small, deep pans and less for a 9X13 pan. (I used a 9X13 pan, and it took 35 minutes.) Test by inserting a toothpick into the center; it’s done when the toothpick comes out clean.
Allow to cool completely before cutting and serving.