1. Preheat grill to 350° to 400° (medium-high) heat. Stir together brown sugar, next 4 ingredients, and 1/2 tsp. salt in a small bowl. Rub chicken with sugar mixture, coating well.
2. Grate zest from limes to equal 1 Tbsp.; squeeze juice from limes to equal 2 Tbsp. Whisk together honey, lime zest, and lime juice in a medium bowl. Add avocado, and toss to coat. Stir in mango, next 3 ingredients, and remaining 3/4 tsp. salt.
3. Grill chicken, covered with grill lid, 5 to 6 minutes on each side or until a meat thermometer inserted into thickest portion registers 165°. Serve with mango-avocado salsa. Garnish, if desired.