Martha likes to garnish individual servings with toasted marshmallows.
Cook first 4 ingredients in a Dutch oven over medium-low heat, stirring occasionally, 15 to 20 minutes or just until milk begins to steam. (Do not boil.) Remove from heat. Remove and discard cinnamon sticks. Whisk in cocoa, sugar, and vanilla, whisking until blended.
Cook over low heat 5 minutes. Add chocolate, whisking until blended. Keep warm over low heat.