In a large pot, bring 4 quarts of water to a boil. Add the udon, stir well and cook until al dente. Drain the noodles, then add 2 cups of ice to the strainer.

Continue running under cool water, tossing, until the noodles are chilled. Drain well, then transfer to a large bowl. Toss with 2 teaspoons of the oil, then set aside.

In a small saucepan over medium, combine the soy sauce, water, mirin and sugar. Bring to a simmer, stirring, then add the dried mushrooms, pushing them into the liquid.

Remove from the heat, cover and set aside until the mushrooms have softened and cooled, 20 to 30 minutes.

Remove the shiitake mushrooms from the soy sauce mixture, squeezing them to allow any liquid to drip back into the pan. Remove and discard the stems, then finely chop. Transfer to a medium bowl and set aside.

In a large nonstick skillet over medium-high, heat 1 tablespoon of the remaining oil. Add the fresh mushrooms and cook, stirring occasionally, until lightly browned and slightly shrunken, about 3 minutes.

Add the onion, drizzle with the remaining 1 tablespoon oil and cook, stirring occasionally, until the onion is softened, about 3 minutes. Stir in the garlic and cook until fragrant, about 30 seconds.

Add the bok choy and cook, stirring, until the leaves are wilted and the stem pieces are crisp-tender, about 2 minutes. Add to the chopped dried shiitakes.

Set the now-empty skillet over medium and add the udon, gently tossing them with tongs. Add the vegetable mixture, gently toss a few times, then add the soy sauce mixture and white pepper. Cook, tossing constantly, until the noodles are heated and have absorbed most of the liquid, about 2 minutes.

Transfer to serving bowls and sprinkle with scallions and sesame seeds. Serve with shichimi togarashi and pickled ginger.