Place potato in a medium saucepan; add water to cover. Bring to a boil; cover, reduce heat, and simmer 12 minutes. Drain and set aside.
Slice each eggplant in half lengthwise; slice halves lengthwise into 1/2-inch-thick strips. Set aside.
Combine broth and next 9 ingredients in a medium bowl; stir well. Pour 1/3 cup broth mixture into a large heavy-duty, zip-top plastic bag. Set aside remaining broth mixture. Add potato, eggplant, chicken, red pepper, and zucchini to bag; seal bag, and shake until chicken and vegetables are well coated. Marinate in refrigerator 3 hours, turning bag occasionally. Remove chicken and vegetables from marinade; discard marinade.
Coat grill rack with cooking spray; place on grill over medium-hot coals (350° to 400°). Place potato on rack; grill, covered, 10 minutes, basting occasionally with reserved broth mixture. Turn potato; place chicken and remaining vegetables on rack. Grill potatoes, chicken, and vegetables, covered, 5 minutes on each side or until chicken is done and vegetables are tender, basting occasionally with reserved broth mixture. Garnish with fresh dill sprigs, if desired.