Make marinade: In a blender, combine scallions, garlic, jalapeno, lime juice, oil, sugar, allspice, thyme, cinnamon, 1 teaspoon salt, and 2 tablespoons water; blend until smooth. Set aside 1/4 cup for brushing.
Place chicken in a shallow dish; season all over with salt. Pour remaining marinade over chicken; toss to coat. Cover; refrigerate, turning once or twice, at least 2 hours, or up to overnight.
Heat grill to medium-high; oil grates. Lift chicken from marinade, letting excess drip off (discard marinade); place on grill, and cover. Cook, turning occasionally, until chicken is blackened in spots, about 10 minutes.
Move chicken to a cooler part of the grill; brush with reserved marinade. Grill, covered, until chicken is cooked through, 10 to 15 minutes more. Serve immediately.