1. Heat oil in a non-reactive medium saucepan. Add onion and cook over medium heat, stirring, until onion is softened, about 3 minutes. Stir in garlic, mustard, and cayenne. Cook 1 minute. Add remaining ingredients except chicken. Bring to a boil; reduce heat to medium-low and simmer, stirring often, until lightly thickened, about 15 minutes. Use immediately or refrigerate up to 2 weeks.
2. Prepare a medium fire in a charcoal or gas grill. Grill chicken, skin side down, 10 minutes. Turn skin side up and grill5 minutes. Brush with some sauce and grill, brushing with additional sauce, until chicken juices run clear when pricked, about 10 to 15 minutes longer. (White meat parts will take less time than legs and thighs.) Boil remaining sauce and serve with grilled chicken.