Quarter the apples and the lemon. Place in a heavy pot with cinnamon sticks. Add apple juice, cider, or water. Cover, bring to a boil, and them simmer over low heat, stirring occasionally to turn the apples and making sure they do not stick. You may want to add some liquid. Cook about 20 minutes, or until the apples are soft. Remove cinnamon sticks. Put the sauce through a food mill and adjust seasoning by adding honey, brown sugar, or maple syrup to taste.