Bring 8 cups unsalted water to a boil in a 6- to 8-quart pot, then add noodles, stirring to separate, and cook 15 seconds. Drain in a colander and rinse under cold water until noodles are cool, then shake colander briskly to drain excess water.
Stir together soy sauce, rice wine, oyster sauce, and cornstarch with 1/4 teaspoon white pepper in a small bowl until smooth, then stir in chicken broth.
Heat wok over high heat until a drop of water evaporates within 1 to 2 seconds of contact, then add 1/2 cup peanut oil and heat until just smoking. Carefully add noodles all at once, flattening top to form a 9-inch cake. Cook until underside is golden, 4 to 5 minutes, rotating noodle cake with a metal spatula to brown evenly and lifting edges occasionally to check color.
Carefully flip noodle cake over with spatula and tongs, then cook, rotating cake, until other side is golden, 2 to 3 minutes more. Transfer noodle cake to a large paper-towel-lined plate to drain excess oil. Discard any oil remaining in wok and wipe out wok with paper towels.
Transfer drained noodle cake to a platter and loosely cover with foil to keep warm.
Heat wok over high heat until a drop of water evaporates within 1 to 2 seconds of contact.
Add 1 tablespoon oil to wok over high heat. When oil just begins to smoke, add ginger and garlic and stir-fry 5 seconds, then add scallions and stir-fry 30 seconds. Add mushrooms and stir-fry until softened, about 3 minutes. Add choy sum and stir-fry until leaves are bright green and just wilted, 2 to 3 minutes (if using broccoli, cook until almost crisp-tender).
Stir broth mixture, then pour into wok and stir-fry until sauce is slightly thickened, about 2 minutes.
Return mixture just to a boil, then pour over noodle cake.