Preheat the oven to 325 degrees. Oil or spray your cooking pan(s).
Puree the beets in a food processor with a bit of water.
Put the pureed beets into a small bowl. Add apple sauce, 1/4 cup water, vanilla extract, and apple cider to the beets and mix well.
In a large bowl, mix the dry ingredients together; then add the beet mixture and stir until well-combined.
Bake for 20-30 minutes (Depending on the size of pan you use: more for small, deep pans and less for a 9x13 pan. Test by inserting a toothpick into the center; it's done when the toothpick comes out clean.).
Allow to cool completely before cutting, frosting, and serving.