In a large mixing bowl, combine beaten eggs with diced red bell pepper, chopped celery, sliced mushrooms, minced onion, bean sprouts, half of the chopped green onions, kosher salt, white pepper, and cooked chicken. Mix gently until well combined.

In a small saucepan over medium heat, whisk together chicken broth, soy sauce, rice vinegar, brown sugar, and sesame oil. Heat until simmering. In a separate bowl, dissolve cornstarch in cold water and stir into the gravy until thickened, about 2-3 minutes.

Heat ½ tablespoon of canola oil in a nonstick skillet over medium heat. Pour in ½ cup of egg mixture, cook for 3-4 minutes until edges lift and bottom is golden brown, then flip and cook for an additional 2-3 minutes.

Transfer cooked omelet to a warm plate, cover with foil. Repeat with remaining oil and egg mixture.

Serve omelets on plates, ladle gravy over them, and sprinkle with remaining chopped green onions.