The chutney can be made up to three days ahead and stored in an airtight container in the refrigerator.
Combine cranberries, pears, orange peel and juice, sugar, cinnamon stick, and 1/2 teaspoon salt in a large saucepan. Bring to a boil, stirring until sugar is dissolved.
Reduce heat to medium-low and simmer, stirring occasionally, until cranberries begin to burst and pears are tender, 8 to 10 minutes. Let cool completely; discard cinnamon stick and serve.