Heat the oil in a medium-sized skillet over medium-high. Add the onions and cook until translucent, about 2 minutes. Add the red peppers, garlic, salt and pepper, and allspice and cook for 2 minutes more. Stir in the vinegar, molasses, Worcestershire sauce, lime juice and tomato sauce and cook for 1 minute more. Transfer the mixture into a bender and add the mango and jalapeno. Blend until smooth.
Put 1 cup of the barbeque sauce into a sealable plastic bag with the chicken and marinate for 1 hour.
Preheat the broiler. Put the chicken on a broiler pan and discard the marinade. Broil the chicken on high for about 12 minutes, turning once. Let the meat rest for 10 minutes before slicing. Spoon about 1/2 cup of barbeque sauce over the meat slices and serve.