Heat a charcoal or gas grill or broiler to moderate heat and put the rack about 6 inches from the heat source. If grilling, keep part of the grill cool for indirect cooking.2. Combine oil, cilantro, lime juice, shallot or onion, cayenne and some salt and pepper. If you're grilling, spread some of the mixture all over the chicken. If you're broiling, spread a little of the mixture on a foil-lined pan; put the chicken pieces on top and sprinkle them with salt and pepper. Spread a little more of the mixture on the chicken. 3. To grill: put the pieces skin side up on the cooler side of the grill. After some of the fat has been rendered, turn the chicken; if it flares up, move it farther away from the fire, or turn it skin side up again. Baste the chicken as it cooks. When the skin has lost its raw look and most of the fat has been rendered, after about 20 minutes, move the chicken directly over the fire. Cook, turning as needed, until both sides are nicely browned and the flesh is firm and cooked through (to check for doneness, cut into a piece close to the bone; the juices should run clear), 5 to 10 minutes longer. To broil: Start with the skin side down and broil, making sure the pieces do not burn, until it's nearly done, about 15 minutes, then turn and cook until done, 5 to 10 minutes longer. Baste the chicken as it cooks. When the chicken is done, drizzle it with the remaining mixture and garnish with additional chopped cilantro. Reserve half the chicken and refrigerate; serve the rest with lime wedges.