Toss the chicken breast, soy sauce, half of the sesame oil, and rice vinegar together in a large bowl. Marinate for at least 20 minutes, up to 24 hours. After marinating, discard the excess liquid.
Mix all of the sauce ingredients in a bowl and set aside.
Heat the other half of the sesame oil over medium-high heat in a large pan. When the oil is heated and begins to shimmer, add the chicken breast and cook until browned on all sides, about 4-5 minutes. Remove chicken from the pan and set aside.
Add additional sesame oil to the pan if it has gone dry. Add mushrooms, broccoli, and carrots to the pan and cook until carrots are tender, about 3-4 minutes. Stir frequently to prevent burning.
Add the onion, green pepper, and red pepper, and cook until the onions are slightly translucent and peppers become slightly tender, about 2-3 minutes.
Add sauce and chicken and continue to cook until the sauce thickens slightly, another 2-3 minutes.
Remove from heat, top with sesame seeds, and serve.