Whisk chicken broth, soy sauce, oyster sauce, rice vinegar, and sugar in a measuring cup; set aside.
In a small bowl, stir cornstarch with water to make a smooth slurry; keep nearby.
Heat a large wok or skillet over high heat. Add vegetable oil. Stir-fry sliced chicken until just cooked through and lightly browned, about 3 to 4 minutes. Transfer to a plate.
Add onion, garlic, and ginger to the pan. Stir-fry until fragrant, about 30 seconds.
Add celery, carrot, bell pepper, and mushrooms. Stir-fry until starting to soften, about 2 minutes.
Add cabbage, snow peas, bamboo shoots, and water chestnuts. Stir-fry 2 minutes more.
Return chicken and any juices to the pan. Pour in the sauce and bring to a simmer.
Stir the cornstarch slurry, then add it to the pan. Toss until the sauce thickens and coats everything, 1 to 2 minutes.
Add bean sprouts and sesame oil; toss just until heated through, 30 to 60 seconds. Season with white pepper and salt.
Serve hot over steamed rice or noodles and garnish with green onions.
Notes
For a vegetarian version, swap chicken for extra-firm tofu and use vegetarian oyster sauce or more soy sauce. Adjust sauce thickness by adding a splash of broth if it gets too thick.