In a small bowl, whisk together 1/4 cup milk and cornstarch to make a slurry. Set aside. Grab a large bowl and fill it with 2 cups of ice and 1 cup of water. Nest another bowl into that bowl so that it's sitting on the ice water. Place a strainer on top of the inner bowl. Set this setup aside.
In a medium saucepan, add remaining milk, cream, 1/2 cup sugar, corn syrup, and salt. Bring to a boil over medium-high heat. Cook for 4 minutes then add in slurry. Continue cooking for 2 minutes or until thickened. Remove from heat.
In a small bowl, add softened cream cheese and whisk to smooth it out. Add about 1/4 cup of heated milk mixture and whisk together until smooth. Add mixture back to rest of milk.
Place cocoa, coffee, and last 1/2 cup of sugar in a small sauce pan. Heat over medium-high heat and cook for 1 minute. Remove from heat and stir in chocolate. Add to milk mixture and stir. Add in vanilla extract and stir.
Pour mixture through mesh sieve into chilled bowl. Stir to cool down the mixture. Place plastic wrap on top of milk mixture to prevent film from forming on top. Chill in refrigerator until really cold, at least 4 hours.
Once chilled, it's time to freeze the ice cream. Freeze ice cream according to your ice cream maker's instructions (mine took about 15 minutes to freeze). In the meantime, make the cinnamon syrup.