Prepare the chicken:
Begin by heating vegetable oil in a wok or a large, deep non-stick skillet over medium-high heat. Once the oil is hot and begins to shimmer, add the chicken pieces. Stir-fry for approximately 5 minutes, or until the chicken is fully cooked through and no longer pink.
Add vegetables and seasonings:
Introduce the chopped onion and sliced carrots to the pan, followed by the chicken broth, oyster sauce, soy sauce, sugar, and sesame oil. Stir everything together and cook over high heat for 5 minutes, allowing the sauce to bubble and the flavors to meld.
Incorporate remaining vegetables:
Add the shiitake mushrooms, red bell pepper, and celery to the pan. Stir-fry for an additional 5 minutes, until the celery becomes tender but still crisp.
Add peas, green onions, and bean sprouts:
Next, stir in the snow peas, green onions, and mung bean sprouts. Cook for another 2 minutes, ensuring that all ingredients are well-combined and heated through.
Thicken the sauce:
To thicken the sauce, dissolve the cornstarch in 1 1/2 teaspoons of water. Add the cornstarch mixture to the pan, stirring constantly. Continue to stir-fry for 1 to 2 minutes until the sauce reaches your desired consistency. Season with salt to taste, then serve the dish with steamed rice.