For the strawberries:
Preheat the oven to 375 degrees F. Combine the strawberries with the sugar in an 8-inch square glass or ceramic baking dish, stirring gently to mix well. Roast for 8 minutes, or until just soft. Let cool slightly.
Puree the berries in a food processor with the lemon juice. Measure 1/2 cup of the pureed berries; refrigerate the rest of the puree for another use.
For the ice cream base:
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.
Combine the remaining milk, the cream, sugar, and corn syrup in a 4-quart saucepan, bring to a rolling boil over medium-high heat, and boil for 4 minutes. Remove from the heat and gradually whisk in the cornstarch slurry. Bring the mixture 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 until smooth. Add the reserved 1/2 cup strawberry puree and the buttermilk and blend well. 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 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.
Recipe courtesy of Jeni's Splendid Ice Creams at Home by Jeni Britton Bauer.