Place 5 1⁄2 cups cold water, the kombu, and dried mushrooms into a sealable container and refrigerate, covered, 8 hours or overnight.

Strain the kombu-mushroom stock into a medium pot and discard the solids. Stir in the tamari and sake and bring the stock to a boil over medium-high heat. Add the fresh mushrooms and carrot and boil until the mushrooms and carrots are tender, about 2 minutes. Stir in the spinach and turn off the heat. Bring a pot of salted water to a boil over medium-high heat. Add the soba noodles and cook, stirring often, until tender, about 8 minutes. Reserve a cup of the soba cooking liquid, then drain the noodles and rinse with cold water. Stir the reserved soba cooking liquid into the pot with the soup.

To serve, divide the soba noodles and soup among four bowls. Garnish each with 1⁄2 teaspoon toasted sesame oil, 1 teaspoon sesame seeds, 1 tablespoon green onions, and 1 egg yolk.

Preheat the oven to 400°F. Cut the broccoli into bite-size florets and cut the stem into 1-inch chunks.

In a large pot, cook the soba according to package directions. Drain, rinse well with cold water, and set aside.

In a small bowl, whisk together the miso, honey, rice vinegar, tamari, and sesame oil.

Place the sweet potatoes on a large, rimmed baking sheet, drizzle with 1 tablespoon of the olive oil, season with salt and pepper, toss to coat, then roast for 20 minutes. Add the broccoli, drizzle with the remaining 1 tablespoon olive oil and about two-thirds of the miso sauce. Toss, season with more salt and pepper, and roast until the broccoli edges are lightly charred and the sweet potato is tender, 12 to 15 minutes longer, tossing once halfway through cooking. Remove from the oven and toss the vegetables to coat.

Meanwhile, in a small saucepan over medium-low heat, warm the broth and ginger.

In a large bowl, combine the soba with the warm broth and the remaining one-third of the miso sauce. Toss to combine and divide among 4 bowls. Top each bowl with one-fourth of the vegetables, garnish with some green onions and sesame seeds, and serve.