Put mushrooms in a heatproof bowl. Bring a medium saucepan filled with water to a boil. Ladle 2 cups water over mushrooms and let sit until very soft, 15 to 20 minutes.
Add noodles to pan and cook until tender, about 6 minutes. Drain and set aside.
Heat 2 tbsp. oil in a wok or large frying pan over high heat. Add brussels sprouts and onion and cook 3 minutes. Drain mushrooms and squeeze out any excess liquid, then add to wok. Cook, stirring occasionally, until onion starts to soften and brussels sprouts are bright green, about 3 minutes. Add garlic and cook until fragrant, about 2 minutes. Stir in seasonings and broth and cook until most of liquid is gone, about 3 minutes.
Add remaining 2 tbsp. oil and the noodles, tossing to coat, and cook until hot, about 2 minutes. Add bean sprouts and cook just until warm. Transfer to a serving dish and sprinkle with green onions. Serve with soy sauce.
Note: Nutritional analysis is per serving.