In a small sauce pot, combine the shiitake mushrooms with enough water to cover. Bring to a boil, remove from the heat, and let stand until softened, 8 to 10 minutes.
While the mushrooms soak, prepare the udon noodles and the eggs.

Bring a medium sauce pot of water to a boil. Add the udon and cook until just tender, 2 to 3 minutes. Using tongs or a spider skimmer, transfer the udon to a colander, reserving the pot of water for the eggs, and rinse the udon with warm water. Set aside.
Return the pot of water to a boil and fill a small bowl with ice water. Carefully lower the eggs into the boiling water and cook for exactly 5 minutes for soft-cooked (for hard-cooked eggs, cook for 3 to 4 minutes longer). Transfer the eggs to the ice water to cool. Carefully peel the eggs and cut in half lengthwise. Wipe out the pot.
While the water heats and the udon and eggs cook, prepare the rest of the meal.

• Cut the tofu crosswise into ½-inch-thick slices.

In a medium frying pan over medium-high heat, warm 2 tablespoons oil until hot but not smoking. Add the tofu and cook, turning once, until starting to brown and turn slightly crisp, 2 to 3 minutes per side. Transfer to a paper-towel-lined plate and season with salt and pepper. Do not clean the pan.
While the tofu cooks, prepare the scallions.

• Trim the root ends from the scallions; thinly slice the scallions.

Transfer the scallions to a small heatproof bowl and season with salt. In the same frying pan used for the tofu, warm 2 tablespoons oil over medium-high heat until hot but not smoking. Pour the oil over the scallions and stir to combine. Set aside.

• Using a slotted spoon or fork, transfer the mushrooms to a cutting board to cool slightly. Discard the stems and soaking liquid; thinly slice the mushroom caps.

In the same pot used for the eggs, warm 1 tablespoon oil over medium heat until hot but not smoking. Add the cabbage, season with salt and pepper, and cook, stirring occasionally, until starting to wilt, 2 to 3 minutes. Stir in the miso noodle soup base and cook until fragrant, about 1 minute. Add the kombu, mushrooms, and 2 cups water. Bring to a boil, reduce to a simmer, and cook, stirring occasionally, until slightly reduced, about 10 minutes. Remove from the heat, remove the kombu, and season to taste with salt and pepper.

Divide the udon between individual bowls, ladle in the soup, and top with the tofu. Garnish with the nori and egg halves. Drizzle with the scallion oil, sprinkle with as much togarashi as you like, and serve.