Preheat oven to 375 degrees F. Hull strawberries and slice. Combine strawberries with sugar in an 8" square glass or ceramic baking dish, stirring gently to mix well. Roast for 8 minutes, or until just soft. Let cool slightly. Puree in a food processor with lemon juice. Measure 1/2 cup of the pureed berries; refrigerate the rest of the puree for another use.
Mix about 2 Tbls 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, 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 back to a boil over medium-high heat and cook, stirring with a heat-proof spatula, until slightly thickened, about 1 minute. Remove from heat.
Gradually whisk the hot milk mixture in to the cream cheese until smooth.Add the buttermilk and reserved strawberry puree 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 mixture into the frozen canister of electric ice cream freezer and run according to directions of ice cream maker. Pack the finished 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. Makes about 1 quart.
Serve with refrigerated strawberry puree poured over the top.