Place raspberries, Stubb’s Smokey Mesquite Bar-B-Q Sauce, honey and vinegar in food processor. Pulse just until combined, leaving some texture and small chunks of raspberries. Add more honey or vinegar to taste. Set aside half the sauce for serving, and use the remainder to baste chicken as it cooks.
Score chicken breasts on both sides in a diamond patter by making parallel cuts about 1 inch apart and 1/8- to 1/4-inch deep. Season on both sides with salt and a good dose of pepper.
Grill over medium-high heat, cooking just long enough to create good grill marks on the first side. Turn chicken and spoon remaining sauce over the cooked side. Grill chicken about 8 to 10 minutes total or until cooked through. Serve hot, passing additional sauce on the side.