In a 3- to 4-quart slow cooker, combine half-and-half, milk, and cinnamon sticks. Cover and cook on low-heat setting for 5 to 6 hours or on high-heat setting for 2-1/2 to 3 hours.
Remove and discard cinnamon sticks. If necessary, skim "skin" from surface; discard. Stir in chocolate pieces; whisk until chocolate is melted and smooth. Stir in vanilla. Serve immediately or keep warm, covered, on warm setting or low-heat setting for up to 2 hours. If desired, float marshmallows on top of individual servings. Makes 12 (3/4-cup) servings.