In a blender or food processor, combine the onion, garlic, thyme, allspice, smoked paprika, cayenne pepper, black pepper, salt, brown sugar, apple cider vinegar, soy sauce, vegetable oil, green onions, Scotch bonnet pepper, and lime juice. Blend until smooth to create the marinade.
Place the chicken thighs and drumsticks in a bowl, pour the marinade over them, ensuring each piece is well coated. Cover with plastic wrap and refrigerate for at least 2 hours, preferably overnight for best flavor.
Preheat your grill or grill pan over medium-high heat. Remove chicken from the marinade, shaking off excess, and discard the leftover marinade.
Once the grill is hot, place the chicken pieces on it, skin side down. Grill for about 6-8 minutes, then flip and grill for an additional 6-8 minutes, or until the internal temperature reaches 165°F (75°C).
Remove chicken from the grill and let rest for 5 minutes before serving.
Serve with sides like rice and peas or grilled vegetables to enhance the meal.