In a food processor, add scallion, garlic, scotch bonnet pepper, ginger, lime juice, soy sauce, vegetable oil, brown sugar, thyme, salt, pepper, allspice, and nutmed. Pulse a few times to break up the big ingredients, then process until consistency is like runny salsa, about 2-4 minutes.
In a large bowl, add sauce and chicken, then toss to coat. Cover bowl and let marinate in the refrigerator for at least 2 hours or ideally up to 24 hours.
When ready to cook, preheat grill to medium-high. If using wooden skewers, prepare them by soaking 6 skewers in water for 20 minutes.
To prepare chicken: thread marinated chicken on a metal or water-soaked wooden skewers. Each skewer should be able to hold anywhere from 6-8 cubes of chicken. Repeat this step until all the chicken is threaded on a skewer. Discard any extra marinade.
Grill assembled skewers until the chicken is filly cooked, about 5-6 minuts per side. Chicken is done when inside is no longer pink and internal temperature is 165 degrees F.