- Prepare the chicken: Remove backbone of chicken with kitchen shears. Split breast bone and crack it so the chicken lays flat. Rub lemon juice over chicken.
- In a small bowl, combine lemon zest, garlic, cumin, paprika, oregano, salt, pepper, and olive oil. Gently loosen chicken skin and rub 3 tablespoons of spice mixture underneath (try not to tear the skin). Rub remaining mixture all over bird and refrigerate at least 4 hours, or overnight.
- Make the green sauce: Prepare a charcoal grill with coals on one side and the fire between 375 and 400 degrees. Grill jalapeño directly over coals, turning after 2 to 3 minutes, or once you see grill marks. Remove from grill and let cool. Remove jalapeño stems (and seeds if you want less heat) and roughly chop. In a food processor, combine jalapeño, garlic, cilantro, lime juice, olive oil, and mayonnaise; blend until smooth. Season with salt and pepper and refrigerate until chicken is ready.
- Grill the chicken: Let chicken sit at room temperature for 20 to 30 minutes. Season with salt and pepper and place on grill skin-side up with legs pointed toward the fire and breast farthest away from the fire. Leave grill vents open to create airflow, which will help crisp the skin. Close lid and cook for 45 to 55 minutes. Flip chicken so it’s skin-side down directly over the coals; cook 10 to 15 minutes more with lid open, watching for flare-ups. Check the internal temperature of a thigh to make sure it’s reached 165 degrees. Remove chicken from grill and let rest 15 minutes.
- Divide chicken into halves with a breast and leg quarter each. Remove leg quarter and separate leg from thigh; cut breasts in half. Serve with green sauce.