Soak mung beans and rice in water overnight. Separately, soak glutinous rice in twice the amount of water overnight.
Rehydrate dried shiitake mushrooms, then dice them along with white dried tofu and dried bamboo shoots. Dice pork and stir-fry until fragrant; set aside. Simmer the diced ingredients in master stock until fully cooked, then set aside. To prepare the stock: combine two bowls of water, one bowl of light soy sauce, half a bowl of dark soy sauce, a small amount of rock sugar, two bay leaves, one star anise, and sliced ginger. Simmer until well-blended. If using new stock, add fatty pork or lard for richness. Steam the soaked glutinous rice over high heat for 20–30 minutes. Alternatively, cook it in a rice cooker with minimal water to avoid softening. While steaming, beat eggs and chop scallions. Reheat the prepared stewed ingredients if made in advance. Blend soaked mung beans and rice into a fine slurry, mix with flour until achieving a thin yogurt-like consistency. Heat oil in a pan, pour a ladle of slurry, and spread thinly. Brush with egg mixture, flip once set, then remove from heat. Spread steamed glutinous rice evenly over the wrapper, top with stewed ingredients and scallions. Cover with a plate, flip the pan, and return to low heat to crisp the egg side. Slide the Doupi onto a plate and serve immediately.