Pat the chicken pieces dry with paper towels and move to a large mixing bowl. In a blender or food processor, combine the peppers, green onions, garlic, ginger, thyme leaves, allspice, nutmeg, a three finger pinch of salt, apple cider vinegar, soy sauce, and brown sugar. Process on low speed until a thick paste forms then stream in the olive oil and process until mostly smooth. Pour the jerk sauce over the chicken massaging in with your hands. Refrigerate the chicken pieces in the sauce overnight.
Make the papaya salad, combine the papaya, onion, green onion, cilantro in a large serving bowl. Add the lemon juice and a hearty pinch of salt and toss with the olive oil. Chill while you cook the chicken.
Prepare a grill or smoker for low to medium heat (we’re aiming for about 300 degrees for the length of cooking). Cook the chicken pieces covered, over indirect heat for 30 to 35 minutes. The chicken is finished when the skin is burnished and the chicken reaches 160 degrees F.
Serve the chicken pieces with the papaya salad.