1Prepare an ice water bath by filling a large bowl halfway with ice and water; set aside. Set a fine-mesh strainer over a large heatproof bowl and set aside.
2Combine cream and milk in a large saucepan and bring to a simmer over medium heat. Remove from heat and whisk in cocoa powder until incorporated. Add chocolate and whisk until completely melted and smooth; set aside.
3Whisk egg yolks in a large bowl until smooth. Gradually add sugar and whisk until mixture is pale yellow and thickened, about 3 minutes. Slowly pour about a quarter of the chocolate mixture into the egg mixture, whisking constantly so the hot mixture doesn’t scramble the eggs.
4Pour chocolate-egg mixture back into the saucepan. Cook over low heat, stirring constantly with a wooden spoon, until the mixture thickens slightly and coats the spoon, about 5 minutes. (When you draw your finger across the spoon, it should make a mark through the mixture, which should not run back in on itself.)
5Remove from heat and strain through the prepared fine-mesh strainer. Stir vanilla extract into the chocolate base, then set the bowl over the ice bath to cool to room temperature, about 15 to 20 minutes.
6Remove the chocolate base from the ice bath, cover, and place in the refrigerator until completely chilled, at least 3 hours or overnight. Freeze in an ice cream maker according to the manufacturer’s instructions. The ice cream will keep in an airtight container in the freezer for up to 1 week.