1. Cook apples, cranberries, sugar, and cinnamon stick in a large heavy saucepan, covered, over medium-low heat, stirring occasionally, until fruit is very tender and broken down into a sauce, about 45 minutes. Discard cinnamon stick. (For a smoother texture, force applesauce through a medium-mesh sieve or a food mill fitted with fine disk into a bowl.) Cool to room temperature or chill before serving.
2. Cooks' note: Applesauce can be made 1 week ahead and chilled, covered.