Preheat oven to 350 degrees.
Pierce beets six to eight times with a fork or knife and wrap in aluminum foil. Place foil-covered beets on a baking sheet in the center of the oven. Bake for an hour or until the beets are tender.
Remove the beets from the oven and allow them to cool fully in the foil. Unwrap the beets and remove the skins with a paper towel. Shred the beets in a food processor.
Place the shredded beets in a saucepan, including all of the juice. Add enough water to cover the beets (about 20 ounces) and cook uncovered over medium heat for 15 minutes. Do not boil.
Add maple syrup, cinnamon, cayenne and mace. Simmer uncovered over low heat for half an hour, or until the liquid coats the back of a spoon.
Remove from the heat and allow the mixture to cool completely.
To make each cocktail, pour sparkling wine or cider into the champagne flutes and slowly add two and a half tablespoons of the spiced beet syrup to each glass. Stir gently.
Drink as is or garnish with the orange peel. The beet syrup will keep in the refrigerator for up to a week.