Rinse dried shiitakes under cold running water to remove any grit. Place shiitakes in a heatproof bowl and pour 1 1/2 cups hot water on top. Soak the shiitakes until fully rehydrated, at least 2 hours or up to overnight. Lift mushrooms from water, squeezing any excess water back into the bowl. Line a fine mesh strainer with paper towels or cheesecloth and strain mushroom soaking liquid through it into a clean bowl and reserve. Discard shiitake mushroom stems.
Place tofu in a clean heatproof bowl and carefully pour enough boiling water on top to cover. Let stand for 30 seconds, then drain and transfer tofu to a paper towel-lined plate. Press gently on tofu with paper towels to dry thoroughly.
Mix together 1/4 cup of cornstarch and 1/4 tsp of salt on a plate. Dredge tofu slices in cornstarch to evenly coat, tapping off excess and transferring them back to a clean, dry plate.
Heat 3 tablespoons oil in a non-stick pan over medium-high heat until shimmering. Working in batches if necessary to avoid crowding the pan, cook tofu slices until crisp and golden on both sides, about 7 minutes. Transfer cooked tofu to a paper towel-lined plate.
Bring a pot of salted water to a boil. Add Chinese mustard greens in and cook until just tender, about 5 minutes. Drain and place greens in a large deep serving plate.
In a bowl, combine vegetable stock, 3/4 cup of the shiitake mushroom soaking liquid, soy sauce, Shaoxing wine, toasted sesame oil, and remaining 5 teaspoons cornstarch. Mix well and set aside.
In a large pan or wok, heat up remaining 2 teaspoons oil. Add shiitake mushroom caps along with all the fresh mushrooms except the enoki, if using. Season with salt and pepper. Cook, stirring, until mushrooms release their water and then it evaporates.
Add vegetable stock mixture and stir well. When the sauce begins to bubble and thicken, gently mix in tofu slices. Lower heat to a gentle simmer, then cook for 5 minutes. Add the enoki mushrooms, if using, and simmer until the mushrooms are tender and coated in a glossy sauce, about 5 minutes longer. Scrape mushrooms and tofu, along with any sauce, on top of mustard greens and serve with white rice.