Mix about 2 tablespoons of the milk with the cornstarch to make a slurry. Set aside. Place chopped chocolate into a large mixing bowl with the vanilla extract and set aside.
Combine the remaining milk, cream, evaporated milk, sugar, corn syrup, and salt in a 4-quart or larger saucepan. Bring to a rolling boil over medium-high heat.
Whisk in cocoa powder to the milk mixture, then continue to boil for 4 minutes.
Remove from heat and whisk in the cornstarch slurry.
Return the ice cream base to heat and continue to cook, whisking constantly, until thick enough to coat the back of a wooden spoon, about 2 minutes.
Pour the hot milk mixture over the chopped chocolate. Allow to sit for about 5 minutes, then whisk to combine. Continue to mix until all the chocolate is melted and thoroughly incorporated.
Chill the ice cream base in the fridge for a minimum of 4 hours or overnight. (To make this faster, you can pour it into a gallon zip-top bag and submerge in an ice bath for about 30 minutes).
Churn the ice cream according to manufacturer's instructions.
Pack the ice cream into your chosen container, layering in the marshmallows. Press a piece of parchment paper directly on top of the ice cream before covering and freezing in the coldest part of your freezer for a minimum of 4 hours.