Soak the dried shiitake mushrooms in cold water for 20 minutes, then drain and slice them into thin strips.
Bring a large saucepan of water to a boil and cook the sweet potato noodles until they are soft but still springy, about 10 minutes. Drain the noodles, then rinse them under cold water. Lay the noodles on a cutting board in a long, straight bundle and cut them into 4- to 5-inch lengths.
Mix the soy sauce, sesame oil, chile paste, sugar and pepper in a bowl.
Heat the canola oil in a skillet or wok over medium-high heat. Add the onion and garlic and cook for 10 seconds. Add the sliced shiitakes and pour in the rice wine, then add the carrot, cabbage, scallions and 2 tablespoons water. Add the spinach and the noodles and toss to heat through. Remove from the heat, stir in the sauce and serve.
Sweet potato thread noodles, also called sweet potato vermicelli, are thin and glassy Korean noodles made from sweet potato starch. Cellophane or bean thread noodles are a fine substitute.