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.
Oxmoor House Cooking Light Collection