Preheat the oven to 300°F. Line a large rimmed baking sheet with parchment paper or a silicone baking mat.
Toss strawberries with 1/4 cup sugar and balsamic vinegar. Spread them in a single layer on the prepared baking sheet and roast for about 30 minutes, until soft and their juices have been released. Mash the strawberries lightly and refrigerate until chilled (about 1 hour).
Blend the ice cream base: In a blender or food processor, puree the cream cheese, whole milk, heavy cream, lemon juice, corn syrup, sea salt, and vanilla extract until smooth. Chill for at least 30 minutes.
Churn the ice cream: Pour the chilled mixture into an ice cream maker and freeze according to the manufacturer’s instructions. In the last 5 minutes, add the roasted strawberries and graham crackers.
Freeze the ice cream: Transfer the ice cream to an airtight container, press plastic wrap against the surface, and freeze for at least 2 hours to firm up.
Serve and enjoy! Once firm, scoop the ice cream into bowls or cones, and enjoy the delicious flavor of strawberry cheesecake in every bite.