In a large mixing bowl, beat the eggs until well combined. Add soy sauce, sesame oil, salt, and pepper, and whisk again.

Fold in bean sprouts, cooked shrimp or tofu, green onions, mushrooms, and carrots. Mix gently but thoroughly to ensure even distribution of ingredients.

Heat vegetable oil in a non-stick skillet over medium heat. Scoop about ½ cup of the egg mixture into the skillet, flattening slightly to form a patty. Cook for 2–3 minutes on each side, or until golden brown and set. Transfer to a plate and repeat with the remaining mixture.

To make the gravy, combine vegetable broth, soy sauce, hoisin sauce (if using), and a pinch of sugar in a small saucepan. Bring to a simmer over medium heat.

Stir in the cornstarch slurry (cornstarch mixed with water) while whisking constantly to avoid lumps. Cook for 1–2 minutes, or until the gravy thickens. Adjust seasoning to taste if needed.

Serve the omelettes hot, topped generously with the warm gravy. Garnish with additional chopped green onions or sesame seeds if desired. Pair with steamed rice, fried rice, or noodles for a complete meal.