In a saucepan, combine the heavy cream, milk, sugar, and cocoa powder. Place the saucepan over medium heat and whisk the mixture until the sugar and cocoa powder are fully dissolved.
Add the chopped chocolate to the saucepan and continue whisking until the chocolate is completely melted and the mixture is smooth. Remove the saucepan from heat.
In a separate bowl, whisk the egg yolks until they are well beaten. Slowly pour about 1/2 cup of the warm chocolate mixture into the egg yolks, whisking constantly to temper the yolks and prevent them from curdling.
Pour the egg yolk mixture back into the saucepan with the remaining chocolate mixture, whisking constantly.
Place the saucepan back over medium heat and cook the mixture, stirring continuously, until it thickens slightly and coats the back of a spoon. This should take about 5-7 minutes. Do not let it boil.
Remove the saucepan from heat and stir in the vanilla extract. Let the mixture cool for about 10 minutes at room temperature.
Transfer the chocolate mixture to a container or bowl, cover it with plastic wrap (make sure the plastic wrap touches the surface of the mixture to prevent a skin from forming), and refrigerate for at least 4 hours or overnight.
Once the mixture is thoroughly chilled, pour it into an ice cream maker and churn according to the manufacturer's instructions. This usually takes about 20-30 minutes.
Transfer the churned ice cream to a lidded container and freeze for an additional 3-4 hours, or until firm.
Serve the chocolate ice cream in bowls or cones, and enjoy!