In a medium saucepan, whisk together cocoa powder, ½ cup of the sugar, and a pinch of salt.
Gradually whisk in the milk and heat over medium heat, stirring until the sugar dissolves and the mixture is warm.
Add the chopped chocolate and stir until melted and smooth.
In a separate bowl, whisk the egg yolks with the remaining ¼ cup sugar until pale and thick.
Slowly pour a little of the warm chocolate mixture into the egg yolks, whisking constantly to temper the eggs.
Pour the tempered yolks back into the saucepan and cook over low heat, stirring constantly until the custard thickens slightly and coats the back of a spoon (170–175°F).
Remove from heat and stir in the heavy cream and vanilla extract.
Strain the mixture through a fine mesh sieve into a bowl to remove any bits of cooked egg.
Cover and refrigerate for at least 4 hours, or overnight, until completely chilled.
Pour the chilled mixture into your ice cream maker and churn according to the manufacturer’s instructions.
Transfer the churned ice cream to a freezer-safe container and freeze for 2–4 hours until firm.
No ice cream maker? Pour the chilled custard into a container, freeze it, and stir every 30 minutes for the first 3 hours to break up ice crystals.