Check your ice cream machine's instructions first. Mine required freezing the bowl 1 day ahead.
Prepare the berry compote: add blueberries, 3 tablespoons sugar, cornstarch, water, and lemon juice to a small saucepan over medium heat. Bring to a boil and then lower heat to low. Cook, stirring often until blueberries are bursting and the compote thickens, about 4-5 minutes. Transfer to a shallow bowl and refrigerate until cooled.
Make the cheesecake ice cream: Combine cream cheese, whole milk, heavy cream, 1 cup granulated sugar, vanilla extract, and a pinch of salt in a food processor. Pulse until well combined. Pour mixture into the bowl of an ice cream maker and churn according to the ice cream maker manufacturer’s instructions.
Once the ice cream is done churning, add half of the ice cream into a freezer-safe container. Add half of the compote on a layer and swirl using a knife. Then add the rest of the ice cream on top, and top with the remaining berry compote and swirl.
Freeze for about 2 hours before serving or overnight.