Rinse the chicken pieces under cold running water, then drain and blot dry with paper towels.
Place the pieces in a non-reactive bowl or baking dish and pour in the lime juice; turn the pieces to coat. Let marinate, at room temperature, for fifteen minutes, turning the pieces once or twice.
Pour off and discard the lime juice, then add the onion, garlic, chili, thyme, oil, paprika, salt, and pepper and turn the chicken pieces to coat thoroughly. Let the chicken marinated in this mixture for at least fifteen minutes or as long as 1 to 2 hours (the longer the better), covering and refrigerating if marinating the longer time.
Preheat the grill, using the two-tiered method (direct and non-direct cooking).
When ready to cook, oil the grill grate. Using a rubber spatula, scrape any bits of onion or garlic off the chicken pieces. Arrange to pieces, skin side down on the grate over the hotter section of the grill and cook for three to five minutes, and finish cooking as directed in the next 2 steps.
After oiling the grill grate, place the pieces, skin side down, on the hotter section of the grill. Cook until the skin starts to brown, three to five minutes. Move the pieces to the cooler section of the grill and continue grilling until the skin is thoroughly browned, five to seven minutes more. Watch carefully and use tongs to move the pieces away from flare-ups.
Turn the pieces and move them back to the hotter section of the grill. Brown the second sides well (three to five minutes), and then move the pieces back to the cooler side of the grill to finish cooking.
The total cooking time will be 16 to 24 minutes. When ready the chicken will be crisp and golden brown outside and juices will run clear when the meat is pierced.