In a small bowl, mix 2 tablespoons of the milk with the cornstarch until smooth to make a slurry.
In a medium bowl, whisk together cream cheese and salt.
In a measuring cup with a spout, mix the cream and corn syrup.
Fill a large bowl with ice and water.
Heat the sugar in a large 4 quart saucepan over medium heat. Watch it carefully and do not walk away. Stand over it with a heatproof spatula but do not touch the sugar until there is a full layer of melted and browning liquid sugar on the bottom with a smaller layer of unmelted white sugar on top. When the edges of the melted sugar begin to darken, use the spatula to help bring them into the center to help melt the white sugar. Continue to stir and push the melted sugar around until completely melted and turns into an even amber color. When little bubbles begin to explode with dark smoke, allow the sugar to cook for another moment and remove it from the heat. Immediately but slowly add about ¼ cup of the cream mixture into the hot sugar, being careful to avoid it popping and spitting at you. Stir until incorporated, then add a bit more cream at a time, stirring, until fully incorporated.
Return the pan to medium high heat and add the milk. Bring to a rolling boil and boil for 4 minutes. Remove from the heat and slowly whisk in the cornstarch slurry.
Bring the mixture back to a boil over medium high heat and cook, stirring with a heatproof spatula, until thickened, about 1 minute. Remove from the heat and, if necessary, pour through a sieve.
Gradually whisk the hot milk mixture into the cream cheese mixture until smooth. Pour the mixture into a 1-gallon ziploc freezer bag, seal, and submerge in the ice bath. Let stand, adding more ice as necessary, until completely chilled, about 30 minutes.
Pour the mixture into your ice cream machine and process according to manufacturer’s instructions. Pack frozen ice cream in a storage container, press a piece of parchment directly against the surface, and seal with an airtight lid. Place in the coldest part of your freezer until firm, at least 4 hours.