In a mixing bowl, beat the eggs until well combined. Add the bean sprouts, green onions, bell pepper, chicken or shrimp (if using), soy sauce, sesame oil, salt, and pepper. Mix until all ingredients are evenly coated with the egg.
Heat 1 tablespoon of vegetable oil in a non-stick skillet over medium heat. Once the oil is hot, pour half of the egg mixture into the skillet, spreading it out evenly.
Cook for about 3-4 minutes, or until the edges start to set and the bottom is golden brown. Carefully flip the omelette using a spatula and cook for another 2-3 minutes on the other side.
Remove the omelette from the skillet and repeat the process with the remaining egg mixture, adding more oil if necessary.
Serve hot, garnished with additional green onions or soy sauce if desired.