Instructions

Make the ice cream:

1. Mix about 2 tablespoons of the milk with the cornstarch in a small bowl to make a smooth slurry. Whisk the cream cheese, peanut butter, and salt in a medium bowl until smooth. Fill a large bowl with ice and water.
2. Combine the remaining milk, the cream, sugar, corn syrup, and honey in a 4-quart saucepan, ring to a roiling boil over medium-high heat, and boil for 4 minutes. Remove from heat, and gradually whisk in the cornstarch slurry. Bring the mixture back to a boil over medium-high heat and cook, stirring with a heatproof spatula, until slightly thickened, about 1 minute. Remove from heat.
3. Gradually whisk the hot milk mixture into the cream cheese mixture until smooth. Pour the mixture into a 1-gallon Ziploc freezer bag and submerge the sealed bag in the ice bath. Let stand, adding more ice as necessary, until cold, about 30 minutes.
4. Pour the ice cream base into the frozen canister and begin to to spin the ice cream. Meanwhile, melt the chocolate in a double boiler. Remove from the heat and let cool until tepid but still fluid. WHen the ice cream is thick and creamy and almost finished, drizzle the melted chocolate slowly through the opening in the top of the ice cream machine and allow it to solidify and break up in the ice cream for about 2 minutes. 
5. Pack the ice cream into a storage container, press a sheet of parchment directly against the surface*, and seal with an airtight lid. Freeze in the coldest part of your freezer until firm, at least 4 hours.
6. *Do this. I don’t know what exactly it does, but it keeps the ice cream that perfect degree of softness that store-bought is when you first open it, even after several days.

Make the swirl:

1. Cook blueberries and sugar in a 12-inch heavy skillet over medium-high heat, thoroughly crushing blueberries with a potato masher, until juices are released and sugar has dissolved, about 2 minutes.
2. Stir together lemon juice and cornstarch, then stir into blueberry mixture. Boil, stirring, 1 minute (mixture will thicken).
3. Transfer blueberry compote to a bowl and chill until cold, about 1 hour.

To finish: When ice cream has finished freezing, transfer to a storage container. Pour blueberry compote over the top and swirl it in. Cover with parchment paper before storing.