Prepare for making your ice cream by placing the ice container in the freezer the day before. Allow it to freeze overnight or for a minimum of 12 hours.
In a medium saucepan, combine the milk, cream, 1/2 cup sugar, cocoa powder, salt, and vanilla. Set over medium heat, stirring occasionally, until the mixture is warm and begins to steam, about 5 minutes. Stir in the melted chocolate.
Combine the egg yolks and the remaining 1/4 cup of sugar in a bowl and whisk until smooth. Gradually incorporate half of the warm milk mixture into the egg yolks, adding one ladleful at a time, until the mixture is warmed. Then, pour the egg-milk mixture back into the saucepan. Cook over medium heat, stirring continuously with a wooden spoon, until the mixture thickens enough to coat the back of the spoon and reaches around 175°F on an instant-read thermometer, typically taking about 5 to 7 minutes. Ensure not to let the mixture boil.
Strain the mixture immediately through a fine strainer into the prepared ice bath. Allow the custard to cool in the ice bath, stirring frequently until it reaches room temperature. Cover the surface of the custard with plastic wrap, then refrigerate until chilled, typically taking about 4 hours or up to 1 day.