Stud the apple with the cloves. Combine the studded apple with the remaining ingredients except the rum and cinnamon in a medium pot. Bring to a simmer over low heat. Simmer until the flavors come together, about 10 minutes. Remove from the heat and add the rum. Discard the apple. Ladle into mugs and garnish each serving with a cinnamon stick.
Makes about 2 1/4 quarts, 8 to 10 servings