In a pan cook frozen blueberry & ¼ cup blueberry preserved over medium heat. Cook until blueberry is soft & the water from blueberry has reduced. The consistency is quite thick.
Turn off the heat. Set aside & put it in a bowl. Let cool for 15 minutes then save it in a fridge.
In the bowl of a blender or food processor, puree the remaining ice cream ingredients except for the graham crackers until smooth. Transfer to a medium bowl and press plastic wrap against the surface. Chill in the refrigerator for at least 30 minutes, or up to 1 day.
Pour the mixture into an ice cream maker. Freeze according to the manufacturer’s directions. During the 5 last minutes of freezing, gradually add the blueberry mixture and graham crackers.
Transfer the ice cream to an airtight container, press plastic wrap against the ice cream, and freeze until firm and flavor are ripened for at least 2 hours.