Make the marinade. Mix lemon juice and zest, olive oil, garlic, tarragon, salt and a generous grind of pepper into a measuring cup or small bowl.
Trim excess fat from chicken, but don’t overdo it. Put chicken into a large zippered plastic bag. Add the marinade to the bag and seal it shut. Slosh it around so the marinade coats the chicken well and marinate in the fridge for at least an hour and up to overnight, turning the bag occasionally to redistribute the marinade.
About 1/2 hour before you’re ready to put the chicken on the grill, take it out of the fridge and let it come to room temperature. If you’re grilling with charcoal, do this when you light the coals.
Arrange the grill for indirect cooking; again, if you’re using charcoal, arrange the coals on one side of the grill. Lightly brush the grilling rack with oil. Remove the chicken from the marinade, shaking off excess marinade, and place it on the grill skin side down away from the coals. Close the lid and cook for 5 minutes. Turn the chicken, close the lid and cook for another 5 minutes.
Move the chicken over the coals or heat, skin side down. Close the lid and cook for 5 minutes. Turn the chicken skin side up; if some pieces are getting charred, swap them around to cooler spots (like where the less charred pieces were). Close the lid and cook for another 5 minutes.
By now, if the chicken’s not done, it should be getting close. Check with an instant read thermometer. If it’s at least 165ºF in the thickest parts, it’s done. If not, move it to the indirect cooking part of the grill, close the lid and continue to cook, checking every 4 minutes or so.
When it’s done, transfer chicken to a platter and let it rest for 5 minutes, then serve.