Make the jerk spice rub by mixing together the allspice, nutmeg, cinnamon, garlic powder, ginger, black pepper and salt together in a small bowl. Set aside.
Holding knife horizontally, insert into the thick end of each chicken breast and cut to make a deep pocket. Be careful not to pierce through the other end.
In a bowl, mix together the cream cheese, mozzarella chesse and ¼ cup Apricot jam until well combined.
Spoon the cheese mixture evenly into the pockets of each chicken breast.
Sprinkle the jerk seasoning all over the stuffed chicken breasts and gently rub spice into the meat.
Make the glaze by blending the remaining ¼ cup apricot jam, scotch bonnet pepper and rum until smooth. Set aside.
Set the Smoker to 121°C (250°F) using Wood Bisquettes of choice (Oak, Maple or Hickory work great).
Place the chicken breasts on smoker rack and put the rack in the Bradley Smoker. Smoke the chicken until they reach an internal temperature of 73°C (165°F), about 4 hours.
Midway through the smoking session, brush some of the glaze on the chicken. Brush them again when the chicken comes out of the smoker.