Stir sugar and cornstarch together in medium saucepan. Gradually stir in milk. Heat, stirring constantly, until sugar dissolves and mixture begins to boil and thicken, about 10 minutes. Add chocolate.

Cook over medium-low heat, stirring, until chocolate is melted and mixture thickens, about 5 minutes. Remove from heat; cool for 5 minutes. Stir in vanilla. Pour into small custard cups. Serve warm or at room temperature. Refrigerate leftovers.