Light the grill: Place a chimney starter on the grate of a charcoal grill. Add newspaper in the bottom, then fill with charcoal. Light the newspaper, which will light the charcoal. Allow charcoal to burn for 30-45 minutes; when the charcoal is mature with ashed-over gray embers, remove the chimney and spread the charcoal around the bottom of the grill. Place the grill plate on top (the coals will be very hot, so use oven mitts).
Harissa paste: In a cast-iron skillet, toast caraway, cumin, coriander, oregano and sesame seeds over medium heat until they're aromatic, stirring occasionally, 1-1½ minutes. Place the toasted spices in a mortar. In the same skillet, toast the peppers over medium heat until they begin to "balloon up," 3–4 minutes. Cut off the stem ends of the chiles and remove the seeds by tapping them out. Tear the chiles into strips and place in a bowl. Cover with hot water, and let stand for 5-10 minutes. Meanwhile, add the peeled garlic and salt to the spice mixture and use the pestle to grind everything together. Stir in the Espelette pepper, followed by the tomato paste. Drain the chiles, squeezing out the water; roughly chop and add to the mortar bowl. Pound and grind until the paste is thick and chunky. Slowly drizzle in the olive oil, enough to just saturate the paste. Add more salt to taste, and a few grinds of black pepper; stir a bit more to combine. Set aside.
Prepare the chicken for grilling: Place the chicken on a rimmed baking sheet fitted with a rack. Tuck the wings back on themselves, then use poultry scissors to cut out the backbone; discard or reserve to make stock. Turn the chicken over, skin side up, and smash the keel bone (breastbone) of the chicken to flatten it out. Flip the chicken skin side down and make a small incision on the top of each inner thigh, which will help the heat penetrate. Dry the chicken thoroughly, then apply the harissa paste on both sides, finishing with the chicken skin side down. Place first skewer through the chicken, beginning at the top part of the thigh, through the bone in the middle of the chicken, to the top part of the breast, finishing just below the wing. Repeat on the other side. (It's important the skewers are fitted below the bone, so they stay in place during grilling.) Refrigerate uncovered, 3 hours to overnight.
Make the garlic confit/butter mop: For the garlic confit, preheat oven to 300 degrees F. In a small saucepan add the garlic cloves, olive oil, lemon peel, and bay leaf, making sure garlic is submerged. Cover and bake until the garlic is tender and golden, 1-1½ hours. For the butter mop, in a small saucepan over medium-high heat, add half of the garlic confit, breaking up the garlic. (The remainder of the confit will last in a jar, refrigerated, for up to two weeks.) Let it sizzle, 2-3 minutes. Add the vinegar and cook until it's dry and the mixture becomes a thick paste. Add butter, 2 tablespoons at a time, whisking with each addition. Bring the butter to a boil, then reduce heat and cover. Whisk in the mustard and a few grinds of black pepper, then set aside until it's time to baste the chicken. (Note: Butter mop can be made a day ahead. Store, covered, at room temperature.)
Grill the chicken: Season the chicken with salt and pepper. Move the pre-heated coals to one side of the grill, then place two bricks on the grate, long side down. Rest the skewers on the bricks so that the chicken is suspended above, but not touching, the grill. This method of indirect cooking will yield a tender, juicy, evenly cooked chicken. Cover and cook 45 minutes, basting with butter mop every 10 minutes.
Finish the chicken: Chicken is done when a thermometer reads 165 F. Alternatively, wiggle the leg; if there is a slight give, the chicken is done. Let rest for several minutes on a cutting board.While chicken is resting, grill lemon halves, cut side down, on the hottest part of the grill. Put greens in a large mixing bowl. Once chicken has rested and the lemons are nicely caramelized, quarter the chicken and cut the breast portions in half. Remove to a serving platter. Squeeze juice from lemon halves directly onto the drippings on the cutting board. Add olive oil and drizzle vinegar on top of the lemon juice to make "Cutting Board Vinaigrette." With a knife or spoon, scrape vinaigrette from the board onto the greens. Toss and serve with the chicken.