Place milk and cinnamon sticks in a medium saucepan. Bring to a boil over medium heat, stirring occasionally. Remove from heat, and let cool 10 minutes. Remove and discard cinnamon sticks.
Add marshmallow cream to milk mixture. Cook over medium heat, stirring constantly with a wire whisk, until thoroughly heated. Stir in chopped chocolate; cook, stirring constantly, until chocolate melts. Remove from heat; stir in vanilla. Pour into individual mugs, and top with whipped cream and shaved chocolate.