Miso Soup with Napa Cabbage and Udon

1 cup sliced dried shiitake mushrooms (about 1/2 ounce)
2 cups boiling water
Kosher salt
1 tablespoon vegetable oil
1/2 medium yellow onion, thinly sliced
1 tablespoon peeled and finely chopped fresh ginger (from about a 2-inch piece)
2 medium garlic cloves, finely chopped
6 cups low-sodium vegetable or chicken stock
2 tablespoons soy sauce
1 (12-ounce) package udon noodles
1/2 medium napa cabbage (about 12 ounces), cored, halved lengthwise, and cut crosswise into 1-inch pieces
1/2 cup white miso
4 large eggs (optional)
 
Place the mushrooms in a medium heatproof bowl and add the boiling water. Let sit until the mushrooms have softened, about 12 minutes.
 
Bring a large pot of heavily salted water to a boil over medium-high heat. Meanwhile, heat the oil in a large saucepan over medium heat until shimmering. Add the onion, ginger, and garlic and cook, stirring occasionally, until the onions have softened, about 5 minutes.

Increase the heat to medium high. Add the stock and soy sauce and stir to combine. Using a slotted spoon, remove the mushrooms from their liquid and add them to the saucepan.  

Measure 1 cup of the mushroom liquid, being careful not to include any sediment from the bottom of the bowl, and add it to the saucepan. Bring to a boil. (Discard the remaining mushroom liquid.)

 
Reduce the heat to low and simmer until the mushrooms are tender, about 15 minutes. Meanwhile, add the udon to the pot of boiling water and cook according to the package directions. Drain in a colander and, while stirring, rinse the noodles with cold water until they’re cooled and no longer sticky. Divide all of the udon among 4 deep soup or noodle bowls; set aside.
 
When the mushrooms are ready, add the cabbage to the pan, stir to combine, and simmer until the cabbage is tender, about 5 minutes. Add the miso and stir to combine. Taste and season with salt as needed. Crack the eggs, if using, into the simmering mixture and cook until the whites are set and the yolks are still runny, about 2 to 3 minutes. Divide the soup and eggs among the bowls of noodles, being careful not to break the egg yolks. Serve immediately.

Serves 4.

From Chowhound.com