Combine milk and sugar in a small saucepan. Heat over medium heat to just a simmer. Then remove from heat.

Stir in cocoa powder using a whisk until well-combined.

Stir in chocolate until melted.

Combine the chocolate and milk mixture in a medium-size bowl with the heavy cream. Stir in the vanilla.

Cover the bowl with plastic and refrigerate for at least 4 hours or overnight if possible.

Churn the ice cream according to your maker's instructions. Freeze ice cream until the desired consistency or eat right away for a soft-serve ice cream.