In a medium bowl, whisk all the sauce ingredients together. Set aside.
For the Stir-Fry:
Add a tablespoon of oil to a large skillet.
Add the chicken and season with a few pinches of salt and pepper. Sauté until the chicken is cooked through, about 5-7 minutes. Remove the chicken from the skillet, and set aside.
Add the rest of the oil to the skillet, then add the onion. Sauté until translucent, or about 5 minutes.
Next, add the broccoli, sugar snap peas, pepper, and carrots to the skillet. Sauté until the vegetables have softened to your liking.
Add the chicken and the sauce to the skillet, stirring to combine the chicken, veggies, and sauce. Simmer the stir-fry until the sauce has thickened (about 3 minutes).
Serve the stir-fry over cooked rice, and garnish (optional) with sesame seeds and green onions.