In a separate bowl, mix the sugar, cocoa powder and cornstarch. Mix until evenly combined and then whisk into the milk mixture.
Place over medium-high heat, stirring occasionally until the mixture starts to steam. While coming up to a warm temperature cut the chocolate into small pieces and set aside. Continue to whisk and bring the ice cream mixture to a boil. When boiling throughout, remove from heat. Pass through strainer if needed.
Whisk in vanilla and chocolate. Stir until the chocolate is completely melted.
Cool off slightly, 5-8 minutes.
Pour into an airtight container and chill 8-10 hours.
To freeze, remove from refrigerator and pour into ice cream freezer. Follow the directions with your freezer; each one can vary. My KitchenAid stand mixer (affiliate link) with ice cream maker attachment(affiliate link) takes about 20-25 minutes to freeze fully.
Dip into airtight container and place in deep freeze an additional 4-6 hours until hardened.