Mix garlic, onion, mustard, brown sugar, olive oil, salt, paprika and pepper in a large food storage bag. Add chicken pieces and seal. Turn to coat. Marinate in refrigerator, turning bag once, about 2 hours.
Prepare grill for medium heat. Remove chicken from marinade, wiping marinade off the pieces, reserve marinade. Grill chicken, turning often, until almost cooked through, about 35 minutes, moving to cooler parts of the grill to avoid burning.
Meanwhile, pour reserved marinade into saucepan and heat to a boil. Reduce heat to medium; cook, stirring occasionally, 10 minutes. Brush marinade over chicken pieces, cook, turning and brushing with marinade, until chicken is cooked through, 15 minutes.