Mix the soy sauce, teriyaki sauce, honey, rice vinegar, minced garlic, and grated ginger in a bowl. Set aside.
Heat one tablespoon of vegetable oil in a large skillet or wok over medium-high heat.
Add the chicken breast slices to the skillet until browned and cooked, about 4-5 minutes per side. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining tablespoon of vegetable oil.
Add the sliced bell peppers, julienned carrot, snap peas, and broccoli florets to the skillet. Stir-fry for 3-4 minutes until the vegetables are crisp-tender.
Return the cooked chicken to the skillet and pour the teriyaki sauce mixture over the chicken and vegetables. Stir well to coat everything evenly.
Cook for 1-2 minutes, until the sauce thickens slightly and coats the chicken and vegetables.
Remove from heat and garnish with sesame seeds and sliced green onions if desired.
Serve the teriyaki chicken stir-fry over cooked rice or noodles.