Broth and reconstituted shitakes can be made and stored in the refrigerator up to 3 days ahead of time, or the freezer up to 3 months. Gently reheat over medium before serving.
Gather dried shitakes in double-lined cheesecloth and tie-off with kitchen twine. Place in a large pot. Add fresh shitake stems, quartered carrots, garlic and cilantro stems.
Toast cinnamon, cloves, star anise, coriander, fennel and black peppercorns in a small skillet over medium-high, stirring, until fragrant, about 2 minutes. Transfer to pot.
Preheat broiler. Place onions, halved shallots and halved ginger on a rimmed baking sheet. Broil, turning occasionally, until very charred in spots, 10 to 15 minutes. Transfer to pot.
Add 16 cups water and season with 2 tablespoons salt. Bring to a simmer over medium-high, skim foam and reduce heat to maintain a low simmer. Cook until broth is reduced by just under half, about 1 hour.
Remove cheesecloth and reserve dried mushroom caps, then pour broth through a fine mesh strainer lined with cheesecloth. Check seasoning; broth should be well-seasoned.
Meanwhile, slice fresh shitake caps into 1/2-inch-thick slices. Heat oil over medium-high in a medium nonstick skillet until shimmering. Add mushroom slices and cook until softened and golden brown, about 7 minutes. Add finely chopped ginger and shallot, and cook, stirring, until fragrant, about 1 minute. Add hoisin and stir to coat. Transfer mushroom slices to a bowl to cool slightly.
Pour boiling water over noodles in a large baking dish or bowl. Let soak 5 minutes, stirring with tongs to release starches, then drain. Repeat, let soak until al dente, about 5 minutes more; drain. Divide among bowls along with reconstituted shitake caps, shitake slices, and julienned carrots. Pour hot broth over noodles and serve with toppings, as desired.