2 tablespoons root beer concentrate or extract, or to taste, depending on its strength
Prep: 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. Fill a large bowl with ice and water.
Cook: Combine the remaining milk, the cream, sugar, and corn syrup in a 4-quart saucepan, bring to a boil over medium-high heat, and 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.
Chill: Gradually whisk the hot milk mixture into the cream cheese until smooth. Add the root beer concentrate. 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.
Freeze: Remove the frozen canister from the freezer, assemble your ice cream machine, and turn it on. Pour the ice cream base into the 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.