Heat several tablespoons of the chicken sauce in a skillet over medium heat. Cook and stir the onion and shiitake mushrooms in the sauce until the onion has softened, about 3 minutes. Add the chicken, green onions, and a few more tablespoons of the sauce; continue cooking until the chicken is no longer pink in the center, about 5 minutes more. Spread the chicken mixture evenly over the skillet; pour the beaten egg overtop. Reduce heat to medium-low and sprinkle with the snow peas. Cook and stir until the egg has firmed and is no longer runny, about 3 minutes.