In medium pot over medium-high heat, bring fruit pulp, barbeque sauce, 1 tsp. chili powder, ½ tsp. Adobo and ½ tsp. ground cumin to boil, stirring occasionally. Reduce heat to medium. Simmer, stirring occasionally, until sauce thickens and reduces to 2 cups, about 10 minutes. Chill.
Using paper towels, pat chicken cutlets dry. In small bowl, mix together remaining 2 tbsp. chili powder, 1½ tsp. Adobo, and ¼ tsp. ground cumin. Sprinkle spice mixture evenly over chicken cutlets.
Heat grill to medium-high heat. Pour 1 cup reserved barbeque sauce into small bowl; reserve remaining cup for serving. Place chicken on hot, greased grill grates. Grill, flipping once, until light golden brown on both sides, about 3 minutes. Using pastry brush, brush top of chicken with mango barbeque sauce. Continue to cook, occasionally flipping and brushing with sauce, until chicken is cooked through, about 5 minutes more.
Transfer chicken to platter. Using clean brush and reserved barbeque sauce, brush chicken with sauce. Serve with remaining sauce.