Heat the oil in a large skillet or wok over medium-high heat. Once the oil is glistening, add the mushrooms, broccoli, baby corn, bell pepper, snap peas, zucchini, and carrots; toss to coat. Cook, stirring often, until vegetables are bright and just tender, 5-6 minutes.
Meanwhile, in a small bowl, combine the soy sauce, sesame oil, garlic, ginger, brown sugar and red pepper flakes, if using. In a separate small bowl, combine the cornstarch with ½ cup cold water. Stir the cornstarch mixture into the sauce.
Pour the sauce over the vegetables and toss to coat. Bring to a simmer and cook until the sauce has thickened and vegetables are tender, about 5 more minutes.
Divide among 6 plates and top with green onions to serve.