Preheat your oven to 300°F. Toss the strawberries with ¼ cup of sugar and balsamic vinegar, then spread them on a baking sheet. Roast for about 30 minutes until softened.
Transfer the roasted strawberries to a bowl and mash gently. Cover and refrigerate until chilled, about 1 hour.
In a blender, combine cream cheese, whole milk, heavy cream, lemon juice, remaining sugar, corn syrup, salt, and vanilla. Blend until smooth.
Chill the mixture in the refrigerator for at least 30 minutes, covered with plastic wrap.
Pour the chilled mixture into your ice cream maker and churn according to the manufacturer’s instructions. Add the roasted strawberries and chopped graham crackers in the last 5 minutes.
Transfer to an airtight container, cover with plastic wrap, and freeze until firm, at least 2 hours.
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Notes
Store in an airtight container in the freezer for up to a month. Let sit at room temperature for a few minutes before serving.