Whisk the eggs with the sugar until light and fluffy (an electric mixer quickens the process). Add milk and cream. Incorporate and add cocoa and vanilla. Mix to dissolve all cocoa. If you wish to add melted chocolate, add it now. This will change the consistency of the ice cream. The melted chocolate will harden into tiny bits giving a grainier result.
Chill mixture for 2-4 hours. This will ensure a better freeze. Follow the manufacturer’s instructions for your ice cream freezer. My KitchenAid attachment takes 20-30 minutes. Once frozen, promptly place in a freezer container and put in the freezer for 2-4 hours for hard-style ice cream. Or enjoy as soft serve right away.