Combine the hoisin sauce and sesame oil in a small bowl.
Whisk together eggs and water in a jug. Heat 2 teaspoons of peanut oil in a wok over high heat. Add one-quarter of egg mixture, tilting wok to cover base and side. As omelette sets, lift edge so that any uncooked egg runs underneath. Cook for 2-3 minutes or until just set. Slide onto a plate and cover. Repeat, in 3 more batches, with 6 teaspoons of oil and the remaining egg mixture, reheating the wok between batches.
Heat remaining peanut oil in wok over medium heat. Add mushroom and cook, stirring, for 2-3 minutes or until tender. Add the gai lan and cook, stirring, for 1-2 minutes or until gai lan just wilts.
Divide omelettes among serving plates. Place one-quarter of bean sprouts on half of each omelette. Top with the mushroom mixture. Fold in half to enclose the filling. Drizzle over hoisin sauce mixture to serve.