To be strictly authentic, you need to acquaint yourself with two ingredients that are common in Peruvian cooking—ají amarillo paste and huacatay (black mint) paste. Both are available at Latin American markets and online. Leave them out if you must. Traditional sides include thick-cut potato or yucca fries and a green salad. But small Peruvian purple potatoes roasted alongside the chicken (dress them with olive oil and salt and pepper before roasting) are also a good complement to the smoky chicken.
Step 1: Remove and discard the fat just inside the neck and body cavities of the chicken as well as the giblets (set aside for another use).
Step 2: In a large deep mixing bowl or stockpot, stir the salt and water until the salt is dissolved. Submerge the chicken in the brine. (Alternatively, use a large resealable plastic bag.) Cover and refrigerate for 6 to 8 hours.
Step 3: Make the wet rub: Place the garlic and salt in a mortar, and using a pestle, pound until a paste forms. (Or use a small food processor.) Work in the paprika, cumin, pepper, ají amarillo paste, huacatay paste, lime juice, and olive oil.
Step 4: Drain the chicken, rinse under cold running water, and pat dry with paper towels. Place the chicken on a rimmed baking sheet. Coat the chicken all over with the wet rub, using your fingers to work some of the rub under the skin of the thighs and breasts. (I like to wear food-safe gloves.) Fold the wing tips behind the back and tie the legs together with butcher’s string.
Step 5: Rotisserie method: Set up the grill for spit-roasting following the manufacturer’s instructions and preheat to medium-high. When ready to cook, thread the chicken onto the rotisserie spit. Attach the spit to the grill and turn on the motor. If using a gas grill, place all of the wood chips in a smoker box or in a smoker pouch and run the grill on high until you see smoke, then reduce the heat to medium-high. If using a charcoal grill, toss half the wood chips directly on the coals. Replenish the coals and chips as needed.
Indirect method: Set up the grill for indirect grilling, place a drip pan in the center, and preheat the grill to medium-high. When ready to cook, brush and oil the grill grate. Place the chicken in the center of the grate over the drip pan and away from the heat and cover the grill.
Step 6: Grill the bird until the skin is golden brown and crisp and the meat is cooked through, 1 to 1 1/4 hours. Use an instant-read meat thermometer to test for doneness; the internal temperature in the thickest part of the thigh should be 165 degrees F. Carefully remove the bird from the spit and let rest, uncovered, for 10 minutes. Serve with Green Sauce.