Dissolve salt in 2 quarts water in large bowl or two 1-gallon zipper-lock plastic bags. Add chicken parts (seal zipper-locks bags, if using), and refrigerate until fully seasoned, about 1-1/4 hours. Remove chicken from brine, rinse very well, dry thoroughly with paper towels, and season with pepper to taste.
Half an hour before cooking, ignite about 5 quarts hardwood charcoal or charcoal briquettes in a chimney and burn until completely covered with thin coating of light gray ash, 20 to 30 minutes. Dump out coals and, spread coals over 2/3 of grill bottom, leaving 1/3 with no coals; position rack over coals and heat for 10 minutes. Coals should be medium-hot (you can hold your hand 5 inches above the grill surface for 4 seconds).
Heat olive oil and garlic in small saucepan over low heat until garlic starts to turn a light brown. Remove from heat, add lemon juice and zest in a large, shallow, nonreactive 13-by-9-inch baking dish or similar pan; set aside. (Dry herbs may be added at this point; fresh herbs should not be added until just before coating chicken.)
Place chicken skin side down on rack directly over to empty side in case of flare-ups. Keep turning and moving pieces to ensure even cooking, until chicken is golden brown, 15 to 17 minutes for legs and wings and 18 to 20 minutes for thighs and breasts. When chicken pieces are well colored, place them in lemon sauce and roll to coat completely. Return pieces to side of grill with no coals; heat until lemon sauce flavors meat, about 5 minutes longer, turning each piece and brushing with sauce once or twice more. Return chicken to pan and roll in lemon sauce once more; serve warm or at room temperature.