Make a pocket in each chicken breast half by cutting horizontally from one side almost to the opposite side. Place chicken in a re-sealable plastic bag set in a shallow dish. For marinade: In a small bowl, combine mustard, mayonnaise, oil, vinegar, oregano, basil, and rosemary. Pour over chicken. Seal bag; turn to coat chicken. Marinate in the refrigerator for 2 to 4 hours, turning bag occasionally. Remove chicken from marinade, scraping off and discarding excess marinade.
In a medium bowl, combine mozzarella cheese, spinach, sweet pepper, garlic, and black pepper. Divide stuffing among pockets in chicken. If necessary, secure the openings with wooden toothpicks.
For a charcoal grill, arrange medium-hot coals around a drip pan in a grill with a cover. Test for medium heat above pan. Place chicken, bone sides down, on the grill rack over pan. Cover and grill for 45 to 55 minutes or until chicken is no longer pink (170 degrees F), turning once halfway through grilling. (For a gas grill, preheat grill. Reduce heat to medium. Adjust for indirect cooking. Cover and grill as above.) Before serving, discard toothpicks. Makes 6 servings.