In a small bowl, mix about 2 tablespoons milk with the cornstarch to make a smooth slurry.
In a slightly larger bowl, whisk the cream cheese and salt until smooth.
Fill a large bowl halfway with ice water.
In a 4-quart saucepan, combine the remaining milk, heavy cream, sugar, and corn syrup and bring to a boil over medium-high heat. Keep it at a boil for 4 minutes.
Remove from the heat and gradually whisk in the cornstarch slurry. Bring the mixture back to a boil over medium-high heat and cook, stirring with a heatproof spatula, until slightly thickened, about 1 minute. Remove from the heat.
Gradually whisk the hot milk mixture into the cream cheese mixture until smooth. Pour the mixture into a 1-gallon resealable plastic freezer bag, squeezing out as much air as possible before sealing it. Submerge the bag filled with the ice cream base in the ice water and let stand, adding more ice as necessary, until cold, about 30 minutes or tuck it in the fridge and let it rest for up to several hours.
Process the ice cream base according to your ice cream machine manufacturer's directions. (The ice cream mixture will be very thick even when you begin to churn it. Don't fret. This is okay.)
Transfer the churned ice cream into a container with a lid. Press a sheet of parchment directly against the surface of the ice cream and seal it. Stash the ice cream in the coldest part of your freezer until firm, at least 4 hours.