Marinate chicken. Add chicken to a resealable container or zip top bag. Pour 1/2 cup of the teriyaki sauce, soy sauce, and rice vinegar over the chicken and mix around with your hands until well incorporated and evenly coated. Set aside to marinate in the refrigerator for at least 2 hours or up to 6 hours.
Set aside 2-3 tbsp teriyaki sauce in a small bowl with a brush for grilling.
Grill chicken. After chicken has marinated, pre-heat grill to 400°. Spray grill grates with grill-safe non-stick spray. Allow excess marinade to drip off of chicken, then place on sprayed or oiled grill grates. Close grill and cook for 6 minutes, then flip chicken, brush with additional teriyaki sauce, and continue cooking for 5-6 more minutes or until internal temperature barely reaches 165°. Set aside to rest for at least 5 minutes before slicing.
Prepare veggies. As the chicken cools, heat a large skillet over medium high (7-8 out of 10) heat. Add 1 tsp sesame oil. Once hot, add julienned carrots and cook, flipping and stirring frequently for 2-3 minutes or until carrots begin to soften. Add pepper and onion and continue cooking for another 1-2 minutes. Remove to a large bowl and set aside.
In the same hot skillet, add remaining sesame oil. Once hot, add broccoli florets and toss to coat evenly. Cover with a lid to trap the steam and continue cooking and flipping broccoli until it's tender crisp, about 3 minutes. Add a tablespoon or 2 of water if needed to help it steam and cook. Remove to the bowl with the cooked carrots, pepper, and onion.
Add shelled edamame to bowl with veggies and pour remaining teriyaki sauce over top. Mix to coat evenly.
Assemble. Serve over rice and add sliced chicken on top. Garnish with sesame seeds, if desired.