Wash and thoroughly dry all vegetables. Cut broccoli into small florets, slice bell pepper into thin strips, trim snap peas, julienne the carrot, and slice shiitake mushrooms. Mince garlic and grate ginger. Have everything ready by the stove – stir-frying moves fast.
In a small bowl, whisk together the soy sauce, rice vinegar, honey (if using), and cornstarch-water slurry. Set aside.
Heat a large wok or heavy skillet over high heat until a drop of water evaporates instantly. Swirl in vegetable oil and immediately add broccoli, carrots, and mushrooms. Stir-fry for 2 minutes, letting them char slightly, then add bell pepper and snap peas. Cook another 90 seconds, tossing constantly.
Push vegetables to the sides, add the sesame oil, garlic, and ginger to the center. Stir for 20 seconds until fragrant. Give the sauce a final whisk, pour it over the vegetables, and toss everything vigorously for 30 seconds until glossy and thickened.
Transfer to a serving platter or individual bowls, sprinkle with sesame seeds and scallions. Enjoy right away while the vegetables retain their crisp-tender texture.