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1. Soak the shiitake mushrooms and lily buds in a bowl of warm water for 15 to 20 minutes. Squeeze out the excess water. Discard the stems from the mushrooms and thinly slice the mushroom caps. (For extra mushroom flavor, reserve the soaking liquid and add it to the dish.) Slice the rough black ends off the lily buds and discard, cut the lily buds in half, and pull apart the strands.
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2. Rinse the bamboo shoots and thinly slice them. Trim and discard the hard ends from the snow peas and cut the snow peas lengthwise. Drain and finely chop the water chestnuts.
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3. In a large bowl, soak the bean thread noodles in enough warm water to cover for 10 minutes to soften them. Drain, shake off the excess water, and set aside.
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4. Drain and rinse the tofu, then pat dry with paper towels. Slice the tofu into 1-inch cubes.
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5. Prepare the sauce: In a small bowl, stir together the vegetable broth, soy sauce, hoisin sauce, sesame oil, and brown sugar. Set aside.
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6. Heat a wok, Dutch oven, or deep 14-inch skillet over medium heat until a bead of water sizzles and evaporates on contact. Add the peanut oil and swirl to coat the bottom. Add the ginger and cook until fragrant, about 30 seconds. Add the mushrooms, lily buds, bamboo shoots, snow peas, water chestnuts, cabbage, and tofu. Add the sauce and the mushroom soaking liquid (if using) and bring to a boil. Reduce the heat to a simmer, cover, and cook for 3 to 4 minutes. Remove the lid, add the drained bean thread noodles, re-cover the wok, and simmer for about another 5 minutes, or until the noodles are cooked. (Don't worry if the dish looks a little soupy; the noodles will absorb the remaining broth.) Ladle everything into a deep serving dish and serve hot.
