Combine 1 cup milk, cocoa, cornstarch, and salt in a large bowl; stir well with a whisk. Set aside.
Cook 2 1/2 cups milk in a large, heavy saucepan over medium-high heat to 180° or until tiny bubbles form around edge (do not boil). Remove from heat; stir in sugar with a whisk until sugar dissolves. Add cocoa mixture to pan, stirring until blended. Bring to a boil over medium heat; cook 2 minutes, stirring constantly.
Combine egg and egg yolk in a bowl, stirring well with a whisk. Gradually add milk mixture to egg mixture, stirring constantly. Return mixture to pan. Cook over medium heat until thick (about 2 minutes); stir constantly. Remove from heat. Stir in chocolate and vanilla; stir until chocolate melts. Serve warm or chilled.