1In food processor or blender, place sauce ingredients. Cover; process until smooth.
2Place chicken breasts in resealable food-storage plastic bag; pour in half of the sauce (cover and refrigerate remaining sauce to serve with cooked chicken and vegetables later). Seal and refrigerate bag of chicken to marinate at least 2 hours but no longer than 8 hours.
3Remove chicken from sauce; discard sauce chicken was marinating in.
4Carefully brush oil on grill rack. Heat gas or charcoal grill. Place chicken on grill over medium-high heat. Cut up vegetables and place on grill pan. Cover grill; cook chicken until juice of chicken is clear when center of thickest part is cut (at least 165°F). Cook vegetables to desired doneness. Serve with remaining sauce.