Whisk the broth, soy sauce, honey, sesame oil, ginger, garlic, cornstarch, and optional red pepper flakes until smooth.
Heat 1 tablespoon oil in a large skillet or wok over medium-high heat. Stir-fry broccoli and carrots for 2 minutes.
Add the remaining oil, mushrooms, and snap peas. Cook for 2 minutes, stirring often.
Add both bell peppers and stir-fry for 1 minute, until the vegetables are almost crisp-tender.
Whisk the sauce again, pour it into the pan, and cook for 1 to 2 minutes until glossy and thickened.
Stir in the green onions and serve immediately over cooked rice or quinoa with sesame seeds if desired.