Heat the vegetable oil in a large saucepan over medium-high heat until it shimmers. Add the fresh shiitake mushrooms and cook, stirring occasionally, until they are browned and tender, about 5 to 7 minutes.

Stir in the water, mirin, rice vinegar, soy sauce, garlic, ginger, dried shiitakes, sesame oil, and chili-garlic sauce. Bring the mixture to a boil, then reduce the heat to medium-low and simmer until the liquid has reduced to about 1 1/2 cups, which should take about 10 to 12 minutes.

While the sauce simmers, bring a large pot of salted water to a boil. Remove and discard the smashed garlic and ginger pieces from the sauce and keep the sauce warm over low heat.

Add the mustard greens to the boiling water and cook until they are just wilted and bright green, about 1 minute. Stir in the fresh udon noodles and continue to cook until the noodles are heated through and tender, about 2 to 3 minutes more.

Drain the noodles and greens thoroughly and return them to the pot. Pour the warm shiitake sauce over the noodles and toss gently to combine. Season with salt and pepper to taste and serve immediately while hot.