In a medium saucepan, combine the milk and sugar. Heat over medium heat until the sugar is dissolved.
Whisk in the cocoa powder until fully incorporated.
Add the chopped dark chocolate and stir until melted and smooth.
Remove the pan from the heat and stir in the vanilla extract.
Allow the mixture to cool to room temperature, then refrigerate until thoroughly chilled.
Once chilled, whip the heavy cream until soft peaks form.
Gently fold the chilled chocolate mixture into the whipped cream until well combined.
Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions.
Transfer the ice cream to a lidded container and freeze for at least 2 hours before serving.