For the daikon broth: Place the daikon, shiitake stems, peppercorns, bay leaves, celery, carrots, cilantro stems, enoki bottoms, onions, red peppers, green onions and 12 cups cold water in a large pot and bring to a boil. Reduce the heat and simmer, about 1 hour. Strain the liquid into a clean pot, season with salt and keep hot.
For the steamed sweet potatoes with green onion vinaigrette: Fill a large pot halfway with water and place a bamboo steamer over top. Bring the water to a boil then reduce to a simmer and steam the sweet potatoes until tender, about 20 minutes.
Heat the oil in a small saute pan. Add the garlic and lightly fry until golden brown, about 5 minutes. Add the rice wine vinegar, fish sauce, chili paste and green onions. Add the cilantro and toss. Spoon the green onion vinaigrette over the steamed sweet potatoes when ready to serve.
For the shiitake cigars: Place the butter and vegetable oil in a saute pan over medium-high heat. Add the garlic and shallots and saute until tender and translucent, about 5 minutes. Add the shiitake caps, season with salt and pepper and saute until golden brown, about 5 minutes. Add the thyme and 1/4 cup daikon broth and stir. Allow the liquid to evaporate, then remove from the heat and place on a dish to cool.
Place 4 cups water and the soy sauce in a wide-mouth saute pan, bring to a boil and then reduce to a simmer. Place a wonton wrapper on a clean, dry work surface and brush with egg wash. Spoon about 1 teaspoon mushroom mixture along the bottom of a wonton. Roll the wonton over tightly, ensuring the mushroom mixture remains in the center and the wonton is sealed. Press down on the edges and then cut to trim excess wonton dough. Repeat with the remaining ingredients. Place the shiitake cigars in the water-soy mixture until the wontons are cooked, about 5 minutes. Remove the shiitake cigars and reserve for the daikon soup.
For the daikon soup: Season the remaining daikon broth with the soy sauce, seasoned rice wine vinegar, fish sauce and honey.
Place 5 shiitake cigars in the bottom of each of 4 soup bowls. Divide the daikon broth among the bowls, garnish with diced daikon, the remaining enoki mushrooms and chopped green onion. Drizzle chili oil over the top. Serve with the steamed sweet potatoes and green onion vinaigrette.