Thoroughly wash the chicken drumsticks, then pat them dry with paper towels. Make shallow cuts on the surface of the meat to help the marinade penetrate.
In a blender, combine the scallions, garlic, grated ginger, chili pepper, soy sauce, brown sugar, allspice, dried thyme, lime juice, and olive oil. Blend until you have a smooth marinade.
Coat the chicken drumsticks with the marinade and let them marinate for at least 2 hours, or preferably overnight in the refrigerator.
Preheat your oven to 400°F (200°C) or prepare a grill over medium heat.
Place the marinated chicken drumsticks on a baking sheet and bake for 40-45 minutes, until golden brown and cooked through. If grilling, grill for 20-25 minutes, turning occasionally, until cooked through.
Serve the jerk chicken with lime wedges and steamed rice or a fresh salad as a side dish.
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