1. For the adobo sauce, whisk lime juice, garlic, cumin, oregano, salt and pepper together in a medium glass bowl. Add chicken; cover and marinate in the refrigerator 1 to 2 hours, turning 2 to 3 times.
2. Heat and oil grill.
3. For the black-bean salsa, fifteen minutes before grilling chicken, combine black beans, mango, red onion, mint, chile, lime juice, olive oil, and brown sugar in a large bowl.
4. Remove chicken from marinade; pat dry with paper towels. Grill chicken 6 minutes; brush with marinade. (Discard remaining marinade.) Turn and grill 4 minutes more, or until done (instant read thermometer inserted near the center registers 170 degree F).
5. Transfer chicken to cutting board and let rest 2 minutes; slice each breast 1/2 inch thick on the diagonal. Arrange chicken breasts and salsa on 4 dinner plates. Serve with rice; garnish with lime slices, if desired. Makes 4 servings.