In a small bowl, whisk together the soy sauce, brown sugar, minced garlic, minced ginger (or ground ginger), and sriracha if using.
In a separate small bowl, mix the cornstarch with 1 tablespoon water until fully dissolved, then whisk this into the sauce.
Heat a large high-sided skillet or wok over medium-high heat and add 1 tablespoon of the sesame oil.
Add the chicken pieces in a single layer and cook 3–4 minutes without stirring to sear, then cook 3–4 more minutes while stirring occasionally until cooked through (165°F internal temperature). Remove chicken to a plate.
Add the remaining 1 tablespoon sesame oil to the pan, then add the zucchini, red bell pepper, shredded carrots, broccoli florets, and diced red onion.
Stir-fry the vegetables, stirring occasionally, for 4–6 minutes until they are tender-crisp.
Return the cooked chicken to the pan, pour in the prepared sauce, and toss to coat everything. Cook 1–2 minutes, stirring, until the sauce has thickened and everything is heated through.
Remove from heat and garnish with chopped green onion and sesame seeds before serving.