Add all teriyaki sauce ingredients to small pot and bring to a boil over medium heat. Lower heat and simmer for 3-5 more minutes until thickened. Set aside to cool.
Marinate chicken in 1/2 of the cooled teriyaki sauce for about 1 hr. Set aside 1-2 tbsp of remaining teriyaki for brushing on during grilling. The rest will be used to drizzle on top before serving.
While chicken marinates, cook rice according to package directions.
Prepare vegetables for grilling by layering two sheets of aluminum foil (approx. 12" x 16") on top of each other. Add vegetables in the middle of the foil and drizzle with sesame oil and soy sauce if desired. Fold up sides into a packet, leaving an opening at the top for steam to escape.
Once chicken has marinated, grill for about 5 minutes per side, depending on thickness and grill heat. Brush with some of the reserved teriyaki sauce when you flip the chicken.
As the chicken grills, add the foil packet of vegetables onto a low flame area of the grill. When the broccoli is bright green and tender crisp, remove the vegetables.
Serve grilled chicken with vegetables, rice and remaining teriyaki.