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1
Place the mushrooms in a medium heatproof bowl and add the boiling water.
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2
Let sit until the mushrooms have softened, about 12 minutes.
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3
Bring a large pot of heavily salted water to a boil over medium-high heat.
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4
Meanwhile, heat the oil in a large saucepan over medium heat until shimmering.
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5
Add the onion, ginger, and garlic and cook, stirring occasionally, until the onions have softened, about 5 minutes.
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6
Increase the heat to medium high.
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7
Add the broth or stock and soy sauce and stir to combine.
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8
Using a slotted spoon, remove the mushrooms from their liquid and add them to the saucepan.
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9
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.
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10
Bring to a boil.
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11
(Discard the remaining mushroom liquid.)
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12
Reduce the heat to low and simmer until the mushrooms are tender, about 15 minutes.
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13
Meanwhile, add the udon to the pot of boiling water and cook according to the package directions.
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14
Drain in a colander and, while stirring, rinse the noodles with cold water until theyre cooled and no longer sticky.
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15
Divide all of the udon among 4 deep soup or noodle bowls; set aside.
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16
When the mushrooms are ready, add the cabbage to the pan, stir to combine, and simmer until the cabbage is tender, about 5 minutes.
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17
Add the miso and stir to combine.
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18
Taste and season with salt as needed.
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19
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.
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20
Divide the soup and eggs among the bowls of noodles, being careful not to break the egg yolks.
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21
Serve immediately, passing togarashi on the side for sprinkling if so desired.
