Combine 2 tablespoons of milk and cornstarch in a small bowl, stirring until smooth. Combine cream cheese and salt in a large bowl, stirring with a whisk. Combine the remaining milk, half-and-half, and the next 3 ingredients (through corn syrup) in a medium saucepan. Scrape seeds from vanilla bean; stir seeds into milk mixture in pan, reserving bean for another use. Bring to a boil; boil for 4 minutes, stirring occasionally.

Add cornstarch mixture; stir with a whisk. Cook, stirring constantly, for 2 minutes or until slightly thick. Gradually add milk mixture to cream cheese mixture, stirring until smooth. Return milk mixture to saucepan. Place pan in a large ice-filled bowl until mixture cools to room temperature (about 10 minutes), stirring occasionally. Pour mixture into the freezer can of an ice-cream freezer; freeze according to manufacturer's instructions.

While ice cream freezes, combine strawberries and 1/4 cup sugar in a small saucepan. Bring to a boil over medium-high heat; cook 5 minutes or until fruit is soft and juice is syrupy. Place pan in a large ice-filled bowl until fruit is cold (about 10 minutes), stirring occasionally.

When ice cream is finished churning, transfer to a container suitable for the freezer, alternating spoonfuls of the ice cream with dollops of the fruit. Press a piece of parchment or wax paper directly on the surface of the ice cream, and freeze until firm, about 4 hours. Remove from the freezer to soften before serving.