Heat coffee and milk in small saucepan over medium heat to a simmer, about 5 min. Place chocolate in large heatproof bowl. Pour hot coffee mixture over chocolate. Whisk until chocolate has melted. Cool completely.
In blender, combine coffee mixture and 2 cups (500 mL) ice cream. Divide into 6 glasses. Top each with scoop of remaining ice cream; garnish with grated chocolate.
Use leftover morning coffee to make these mocha milkshakes as a treat later in the day.