Prep
For the chocolate syrup: Combine cocoa, coffee, and sugar in a small saucepan, bring to a boil over medium heat, stirring to dissolve sugar, and boil for 30 seconds. Remove from heat, add chocolate, and let stand for 5 minutes. Stir the syrup until smooth. Set aside.
For the ice cream base: Mix with a fork about 2 tablespoons of milk with cornstarch in a small bowl to make smooth slurry. Whisk cream cheese or crème fraiche, warm chocolate syrup, and salt in a large bowl until smooth. Don’t worry if mixture sticks to the whisk.
Fill a large bowl with ice and water.

Cook
Combine remaining milk, cream, sugar, and corn syrup in a 4 or 5-quart saucepan, bring to a rolling boil over medium-high heat, and boil for 4 minutes. (Watch the pan so mixture doesn’t boil over.)
Remove from heat and gradually whisk in cornstarch slurry.
Bring mixture back to a boil over medium-high heat and cook, stirring with a heatproof spatula, until slightly thickened, about 1 minute.
Remove from heat.
Gradually whisk hot milk mixture into cream cheese mixture until smooth (it may be easier to transfer the hot milk mixture to a 4-cup measuring cup if your saucepan is heavy).

Chill
Pour mixture into a 1-gallon Ziploc freezer bag and submerge the sealed bag in the ice bath.
Let stand, adding more ice as necessary, until cold, about 30 minutes.
Alternately, if you are not finishing the ice cream on the same day you can chill the mixture in refrigerator for up to 2 days.

Freeze
Pour the ice cream mixture into the frozen canister of your ice cream maker and spin until thick and creamy.
The ice cream is finished at the exact moment when the machine isn’t freezing the ice cream anymore; ice cream will begin to pull away from the side (about 20-25 minutes).
Pack ice cream into a freezer-safe storage container, press a sheet of parchment paper directly against the surface, and seal with an airtight lid.
Freeze in the coldest part of your freezer until firm, at least 4 hours and up to a few weeks.

Adding Flavoring
If adding flavoring such as peppermint or orange, add 1/4 teaspoon peppermint or orange oil just as you turn on the ice cream maker (after you’ve added the ice cream mixture).