Combine the milk (except for 2 tablespoons), cream, coffee, sugar, and corn syrup in a saucepan and bring to a boil on medium-high heat for about 5 minutes. Remove from heat. Allow to steep for about 10 minutes.
While the milk is steeping in the coffee, combine the remaining two tablespoons of milk and the cornstarch together and mix well in a small bowl. In a medium bowl, beat the cream cheese and rested until smooth. Stir the cornstarch mixture into the cream cheese until fully incorporated with as few lumps as possible.
Strain the coffee out of the milk mixture using a cheesecloth to remove all the coffee grounds.
Slowly drizzle some of the hot milk mixture into the cream cheese mixture and mix well until fully incorporated. Pour the cream cheese mixture back into the milk mixture and place the liquid back into the saucepan.
Stir well and simmer until it is thick and creamy. Remove from heat and place in a gallon freezer bag. Add in the vanilla and mix in the bag. Place the bag in an ice bath for about 15 minutes, then place in the refrigerator until the mixture is completely cold—at least 40 degrees F.
Churn according to your ice cream machine's manufacturer instructions. Add in the chopped cookies during the last five minutes of churning. Place in an airtight container and freeze solid for at least 4 hours.