Add the vegetable oil to a large sauté pan or wok set over high heat. Once the oil is smoking, add the sliced chicken and cook it until it is browned and cooked through.
Push the chicken to one side of the pan, then add the chilies (optional), broccoli, bell pepper and onion. Stir-fry the ingredients together until the onion is translucent and the broccoli is fork-tender.
Push all the ingredients to one side of the pan and add the beaten eggs. Lightly scramble, stirring, until they are cooked through, then push all of the ingredients to one side of the pan and add the noodles.
Cook the noodles, stirring constantly, until they are slightly browned on the edges then add the sauce and basil and stir together until sauce heats through and thickens slightly. Garnish with scallions and serve immediately.