In a small saucepan on medium heat, stir together the frozen cherries, tomato paste, balsamic vinegar, light brown sugar, water, Worcestershire sauce, and spices. Heat until the cherries become tender and plump, about 5 minutes.
Using an immersion blender, blend the sauce mixture until smooth. If you don’t have an immersion blender, transfer the mixture to a heat-safe countertop blender, blend until smooth, and then transfer back to the saucepan.
Simmer the blended sauce on low for 15 to 20 minutes until reduced. Remove from the heat and let cool. Once cool, transfer to a jar or airtight container.
Season the chicken drumsticks with salt and pepper. Cook the drumsticks either on the grill or in the oven. Just before the drumsticks are done, brush them generously with cherry BBQ sauce.
Once cooked, remove the drumsticks from the heat. Brush again with cherry BBQ sauce and serve. Store any leftover cherry BBQ sauce in a jar or airtight container in the fridge.