Roast the Strawberries: Preheat oven to 375 degrees F.
Combine the strawberries with the sugar in an 8-inch square baking dish. Roast for 8 minutes, or until barely soft. Let cool slightly.
Place the strawberries in a food processor, along with the lemon juice. Puree until smooth. Place a fine-mesh sieve over a large measuring cup and strain the strawberry puree. Set aside ½ cup for the recipe; refrigerate the remaining for another use.
Make the Ice Cream Base: Mix 2 tablespoons of the whole milk with the cornstarch in a small bowl to make a smooth slurry; set aside.
Whisk the cream cheese and salt in a medium bowl until smooth. Fill a large bowl with ice and water; set aside.
Combine the remaining whole milk, the cream, sugar and corn syrup in a 4-quart saucepan and bring to a rolling boil over medium-high heat. Boil for 4 minutes. Remove from the heat and gradually whisk in the cornstarch slurry. Return the pan to medium-high heat and bring to a boil. Cook, stirring constantly with a rubber 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 ½-cup strawberry puree and the buttermilk and whisk to thoroughly combine. Pour the mixture into a 1-gallon resealable freezer bag and submerge the sealed bag in the ice bath. Let stand, adding more ice if necessary, until cold, about 30 minutes.
Churn the Ice Cream: Churn the ice cream in your ice cream maker according to the manufacturer's directions. Place in an airtight container and freeze for at least 4 hours before serving.