Heat the vegetable oil in a Dutch oven over medium-high heat until it shimmers. Add the fresh mushrooms and cook, stirring occasionally, until they are softened and lightly browned, about 5 minutes.

Stir in the water, mirin, vinegar, soy sauce, garlic, ginger, dried mushrooms, sesame oil, and chili-garlic sauce. Bring the mixture to a simmer, then reduce the heat to medium-low and cook until the liquid has reduced by half, 8 to 10 minutes.

Remove the garlic and ginger pieces from the sauce and discard. Cover the pot to keep the sauce warm and set it aside off the heat.

Bring 4 quarts of water to a boil in a large pot. Add the greens and salt, and cook until the greens are almost tender, about 4 to 5 minutes.

Add the noodles to the pot with the greens and cook until both are tender, about 2 minutes. Reserve 1/3 cup of the cooking water, then drain the noodles and greens and return them to their pot.

Pour the prepared mushroom sauce and the reserved cooking water over the noodles. Cook over medium-low heat, tossing constantly, until the sauce thickens and clings to the noodles, about 1 minute.

Season with salt and pepper to taste and serve immediately while the noodles are hot and glossy.