In a bowl, stir together 1⁄4 cup milk and the cornstarch; set slurry aside. In a 4-qt. saucepan, whisk together remaining milk and the cream, sugar, syrup, and salt; bring to a boil over medium-high heat. Cook for 4 minutes; stir in slurry. Return to a boil and cook, stirring, until thickened, about 2 minutes.
The next day, puree your jam with the ice cream base using an immersion blender. Pour the mixture into an ice cream maker and process according to manufacturer's instructions. I did this in two batches as my ice cream maker has a 500 ml limit. Transfer ice cream to a storage container and freeze until set.
Store the ice cream in a sealable plastic container.