In small saucepan set over medium-high heat, combine blueberries, onion, sugar, apple cider vinegar, and allspice; bring to a boil. Reduce heat to medium; cook, stirring often, for 8 to 10 minutes or until thickened to syrupy consistency. Let cool completely.
Grill patties according to package directions. Serve in buns topped with Blueberry Ketchup, Brie, arugula, red onion, and radishes.
Substitute turkey or chicken burgers for beef patties if desired
If using frozen blueberries, rinse before using.