Make the gravy: in a small saucepan combine 1/2 cup chicken stock, 1 tablespoon oyster sauce, 1/4 teaspoon sugar, and 1 teaspoon light soy sauce. Heat over medium until the mixture is warm and begins to bubble gently.
Prepare the cornstarch slurry: in a small bowl whisk together 1 1/2 teaspoons cornstarch and 4 teaspoons water until smooth.
Add the slurry to the bubbling gravy, stirring constantly until the sauce thickens. Turn off the heat and leave the gravy covered on the warm burner or set aside to keep warm.
Make the egg mixture: in a large bowl beat 6 large eggs until the whites are fully broken down. Stir in 1/2 cup cooked chicken or shrimp, 1/2 cup thinly sliced celery, 1/2 cup thinly sliced mushrooms, 1/2 cup thinly sliced green onion whites, 1 teaspoon soy sauce, and 1/4 teaspoon sesame oil. Mix until evenly combined.
Heat a wok or large nonstick skillet over medium-high heat. Add about 1 tablespoon vegetable oil and swirl to coat the base and lower sides.
Pour about 1/2 cup of the egg mixture into the hot oil. The eggs should bubble on contact. Gently push uncooked egg from the center toward the edges so the uncooked portion can reach the hot surface; let cook until the bottom is set, about 2 minutes.
Carefully flip the egg foo young and cook the other side until golden brown and cooked through, about 2 to 3 minutes more. Transfer to a plate and keep warm. Add more oil to the pan as needed and repeat with remaining batter.
To serve, reheat the gravy if it has cooled, spoon the warm gravy over the egg foo young, and top each portion with the 1/4 cup thinly sliced green onion tops.