Place rice noodles in a bowl and fill with hot water, separating the noodles with a fork or your hands; soak until softened, about 45 minutes.
Whisk oyster sauce, white sugar, vinegar, and lime juice together in a bowl until sauce is smooth.
Pour enough oil into a wok or large skillet to coat the bottom and turn to medium-high heat. Cook and stir onion, shallot, and garlic in hot oil until fragrant and slightly softened, 2 to 3 minutes. Add chicken and bell pepper; cook and stir until chicken is no longer pink, 5 to 7 minutes. Move chicken mixture to 1 side of wok.
Pour egg into the hot wok; cook until it becomes slightly set, 1 to 2 minutes. Flip egg and cook until set, 1 to 2 minutes more. Break egg apart with a fork and stir into chicken mixture.
Mix basil and sauce into chicken mixture. Drain noodles and add to chicken mixture; cook and stir until heated through, 2 to 3 minutes.