Whisk together the oil, 2 tablespoons of the garlic, 2 tablespoons of the thyme, 1 tablespoon of the lemon zest and 1 tablespoon of onion in a large baking dish. Add the chicken and turn to coat in the oil. Cover and let marinate in the refrigerator for at least 2 hours and up to 8 hours.
Preheat the grill to medium. Remove the chicken from the marinade and season both sides with salt and pepper. Place the chicken on the grill, skin side down and slowly grill until the fat renders and the skin becomes golden brown and crisp, about 15 minutes. Turn the chicken over, close the cover and continue grilling until just cooked through, about 20 minutes longer. Remove from the grill, loosely tent with foil and let rest 10 minutes before cutting.
Mix together the remaining garlic, thyme and 1 tablespoon of lemon zest and sprinkle over the cut chicken.