How to Make It
Make the chicken: Rub oil all over chicken. Season chicken, including cavity, with salt and pepper. Stuff cavity with onion quarters, lemon halves, thyme sprigs, and bay leaves.
Place chicken on a work surface, breast side up, with the legs pointing toward you. Center a 3-foot piece of butcher’s twine under the chicken and bring both sides around the wings (securing them to the carcass). Criss-cross twine over the breasts (forming an “X” on breasts) and around the legs (securing them together). Tie to secure. Set aside at room temperature for at least 30 minutes or up to 1 hour.
Make the salsa: Preheat grill to high (450°F to 500°F) on one side, or push hot coals to one side of a charcoal grill. Toss together tomatoes, onion, garlic, and oil in a large bowl. Place a grill basket over the fire and preheat for 5 minutes. Add tomato mixture to the basket, and cook, covered, stirring or shaking basket occasionally, until tomatoes are blistered on all sides and onion is charred and softened and garlic is blackened and feels soft when squeezed, 10 to 12 minutes. Transfer each item to a bowl as it’s ready. Add chiles to basket. and toast, shaking basket constantly, until chiles are fragrant and a shade darker, 30 seconds to 1 minute. Place chiles in hot water; let soften for 10 minutes.
Reduce grill temperature to medium-low (325°F and 350°F). Place chicken, breast side up and legs closest to the coals, over indirect heat. Grill, covered, maintaining internal temperature between 325°F and 350°F, for 35 minutes. (This is a good time to refill and light your charcoal chimney so you have hot coals ready if you think you’ll need them.)
Meanwhile, drain, stem, and seed the chiles, then coarsely chop. Peel garlic, and place it in the bowl of a food processor with chiles, tomatoes, onions, and oregano. Pulse until finely chopped, about 8 times. Add vinegar; honey, if using; and salt to taste; pulse to combine.
After 35 minutes, use tongs to carefully rotate the chicken 180° (so neck is facing heat) and continue to cook, covered, maintaining internal temperature between 325°F and 350°F, until a thermometer inserted in the thickest part of the chicken leg reads 165°F, 40 to 50 minutes.
Transfer the chicken to a cutting board; let rest for at least 15 minutes before carving. Carve the chicken as desired, and serve immediately with the salsa, tortillas, cabbage, radishes, cheese, and limes.