Soak the noodles in room temperature water overnight or in warm water for at least 1 hour. Soak the dried shiitake mushrooms overnight in about 1½ cups of water. If preparing the mushrooms the day of cooking, you can use boiling water to soak the mushrooms for 2-3 hours. Reserve the mushroom soaking water. Thinly slice the mushrooms (you should have about ½ cup), and rip/cut noodles in half to make them more manageable in the wok.
Beat the eggs with the salt, ½ teaspoon Shaoxing wine, and sesame oil. Pre-mix the sauce by mixing ½ cup of the reserved mushroom soaking water, light soy sauce, oyster sauce, sugar, dark soy sauce, sesame oil, and white pepper.
Preheat your wok over medium heat until it’s lightly smoking, or preheat a skillet until a drop of water beads on the surface. Add 1 tablespoon of neutral oil to the wok and swirl to coat the pan. Scramble the eggs–they should puff up and cook quickly. Cook until they’re just set and transfer back to the bowl.
Add the remaining tablespoon of oil to the wok, along with the lap cheong and mushrooms. Cook for 3-4 minutes, until the mushrooms are lightly caramelized.
Add the garlic, carrot, red onion, and the white parts of the scallions. Stir-fry for 2 minutes. Increase the heat to high. Add the remaining tablespoon of Shaoxing wine around the hot perimeter of the wok. Then add the noodles. Toss with your wok spatula to distribute the vegetables and noodles evenly.
Add the prepared sauce mixture and toss the noodles some more. There will be a good amount of liquid in the wok at first, but it will help soften the noodles and soak in. Stir-fry for 3 minutes over high heat until there’s no standing sauce. Add the green parts of the scallions and cook until they’re just wilted. Serve!