In a large, sealable freezer bag or container with lid, mix water, salt, and sugar. Add chicken parts and seal container or bag. Refrigerate for 1 hour and up to 6 hours.

Whisk together one of the vinaigrette ingredient combinations in a large bowl, and set aside.

If using a gas grill, heat all burners to high for 10 minutes, then adjust to moderately high right before you add the chicken. If using a charcoal grill, leave about one-quarter of grill free of charcoal and heat the rest of the charcoals until they’re grayish-white, about 15 minutes.

Lightly oil your grill racks. Arrange chicken on racks, cover with lid, and cook until well-browned, turning over once, about 6 to 8 minutes total for smaller parts (wings, thighs, and drumsticks) and 8 to 10 minutes for breasts.

Once chicken is well-browned, if you’re using a gas grill with multiple sections, turn off the center heat and move chicken pieces onto it. If you’re using a charcoal grill and have left an area free of charcoal, move the chicken onto it. If you’re using a gas grill with one heat control, reduce it to medium.

Cook browned chicken, covered with lid, moving chicken around grill as needed and turning over occasionally, until cooked through, anywhere from 12 to 20 minutes or until a thermometer inserted into the deepest part of your piece of chicken is 160 to 165 degrees.

When chicken is almost done, place lemon or lime halves, cut sides down, over lit burner until grill marks appear, about 2 to 3 minutes.

Transfer chicken to bowl with vinaigrette and toss to evenly coat. You can also cover this bowl with foil to keep it warm until needed. Serve grilled chicken with grilled lemons or limes, and any extra vinaigrette on the side.