1. In a small bowl, combine thyme, allspice, cayenne, 1 tablespoon sugar, 1 teaspoon salt, and 1/2 teaspoon pepper; set spice mixture aside. In a medium bowl, stir together pineapple, lime juice, scallions, and remaining 2 tablespoons sugar. Season stuffing with salt and pepper, and set aside.

2. Lay chicken flat on a work surface. With a paring knife, cut a 2-inch-long slit in the thick side of each breast; insert knife, and pivot inside chicken without enlarging opening, carefully forming a deep pocket (make sure not to pierce the opposite side). Dividing evenly, tightly pack each breast with stuffing; rub exterior with oil, and sprinkle with spice mixture.

3. Heat broiler, with rack set 4 inches from heat. Line a large rimmed baking sheet with aluminum foil; arrange chicken on sheet. Broil until chicken is opaque throughout and an instant-read thermometer inserted into stuffing reads 165 degrees, 15 to 20 minutes. Serve stuffed chicken with rice.