Danger! This is the dry-burn technique. I don't add water to the sugar before putting it on the heat, as some chefs do. Caramelizing sugar dry means it goes faster, but you have to watch it more closely and be ready with your cream. Here is an overview of what you are going to do:
Stand over the pan of sugar with a heatproof spatula ready, 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 the top. When the edges of the melted sugar begin to darken, use the spatula to bring them into the center to help melt the unmelted sugar. Continue stirring and pushing the sugar around until it is all melted and evenly amber in color--like an old penny. When little bubbles begin to explode with dark smoke, give the sugar another moment and then remove from the heat. Immediately but slowly pour about ¼ cup of the cream and corn syrup mixture into the burning-hot sugar. Be careful! It will pop and spit! Stir until it is incorporated, then add a bit more cream and stir, then continue until it is all in.
Mix about 2 tablespoons of the milk with the cornstarch in a small bowl to make a smooth slurry.
Whisk the cream cheese and salt in a medium bowl until smooth.
Mix the cream with the corn syrup in a measuring cup with a spout.
Fill a large bowl with ice and water.
Heat the sugar in a 4-quart saucepan over medium heat until it is melted and golden amber in color (see note above). Remove from the heat and, stirring constantly, slowly add a bit of the cream and corn syrup mixture to the caramel: It will fizzle, pop, and spurt. Stir until well combined, then add a little more and stir. Keep adding the cream a little at a time until all of it is 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 gradually whisk in the cornstarch slurry.
Bring back to a boil over medium-high and cook, stirring with a heatproof spatula, until slightly thickened, about 1 minute. Remove from the heat. If any caramel flecks remain, pour the mixture through a sieve.
Gradually whisk the hot milk mixture into the cream cheese until smooth. Pour the mixture into a 1-gallon Ziploc freezer bag and submerge the sealed bag in the ice bath. Let stand, adding more ice as necessary, until cold, about 30 minutes.
Pour into frozen canister and spin until thick and creamy.
Pack the ice cream into a storage container, press a sheet of parchment directly against the surface, and seal with an airtight lid.
Freeze in the coldest part of your freezer until firm, at least 4 hours.