Make the ice cream: In a bowl, stir together 1?4 cup milk and the cornstarch; set slurry aside. In a 4-qt. saucepan, whisk together remaining milk and the cream, sugar, syrup, and salt; bring to a boil over medium-high heat. Cook for 4 minutes; stir in slurry. Return to a boil and cook, stirring, until thickened, about 2 minutes. Place cream cheese in a bowl and pour in 1?4 cup hot milk mixture; whisk until smooth. Then whisk in remaining milk mixture.

Make the chocolate sauce: Bring cocoa, brewed coffee, and sugar to a boil in a 2-qt. saucepan over high heat; cook for 30 seconds. Remove from heat and stir in chocolate.

Stir sauce into ice cream base. Pour mixture into a plastic bag; seal, and submerge in a bowl of ice water until chilled. Pour mixture into an ice cream maker; process according to manufacturers instructions. Transfer ice cream to a storage container and freeze until set.