1. Preheat the oven to 375F. Combine the strawberries with sugar in a small baking dish and roast for about 8 minutes, or until soft. Allow to cool. Puree the berries with the lemon juice in a food processor.

2. For the ice cream: mix 2 tablespoons of the milk with the cornstarch in a small bowl until smooth. Whisk the cream cheese and salt in a medium bowl until smooth.

3. 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 whisk in the cornstarch. Then return the mixture to a boil over medium-high heat and cook until thickened about 1 minute. Remove from the heat.

4. Stir the hot milk mixture into the cream cheese until smooth. Add ½ cup of the strawberry puree to the mixture. Save the rest of the strawberry puree for later. Chill the mixture in the bowl in the fridge for at least 2 hours.

5. Churn the ice cream in a maker according the manufacturers instructions. Add the chunks of shortcake halfway through churning. Freeze the ice cream for 2 hours before serving.