How to Make It
In a small heatproof bowl, cover the shiitake mushrooms with boiling water. Let stand at room temperature until softened, about 30 minutes. Drain well, discard the stems and cut the caps into 1/2-inch strips.
Using a sharp knife, slice the tofu crosswise 1/4 inch thick. Arrange the slices on a rimmed baking sheet lined with paper towels. Cover with more paper towels and set a second baking sheet on top. Weigh it down with a cast-iron skillet for 30 minutes. Pat the pressed tofu dry with paper towels and cut into 1-inch squares.
In a large heatproof bowl, cover the noodles with boiling water. Let stand until pliable, 10 to 15 minutes. Drain, rinse under cold running water and drain again. Spread the noodles on a rimmed baking sheet, gently pulling the noodles apart.
In a small bowl, whisk 1 tablespoon of the cornstarch with the low-sodium soy sauce, rice vinegar and brown sugar to make a smooth stir-fry sauce.
In a pie plate, use a fork to blend the remaining 1/4 cup of cornstarch with 1 1/2 teaspoons of salt. Working in batches, gently toss the tofu in the cornstarch until lightly coated. Arrange the tofu on a large plate or rimmed baking sheet in a single layer.
In an extra-large skillet or a wok, heat 1/4 cup of the oil over high heat until shimmering. Add half of the tofu and cook, stirring occasionally, until lightly browned all over, about 3 minutes. Using a slotted spoon, transfer the tofu to a paper towel–lined plate. Repeat with the remaining tofu. Discard the oil and wipe out the skillet.
In the same skillet, heat the remaining 2 tablespoons of the oil over high heat until shimmering. Add the broccoli, carrots, cremini mushrooms and a pinch of salt and stir-fry until the vegetables are crisp-tender, about 3 minutes. Add the shiitake mushrooms, bok choy, snap peas, garlic and ginger and cook until the bok choy has just wilted and the snap peas are crisp-tender, about 2 minutes. Reduce the heat to moderately high, add the stir-fry sauce and tofu and stir briskly until the vegetables are coated. Add the noodles in batches, stirring well after each addition. Remove the skillet from the heat and stir in the wine, dark soy sauce and half of the scallions. Garnish with the remaining scallions and peanuts and serve.