Bring broth, vinegar, mirin, mushrooms, ginger, chili sauce, soy sauce, and sesame oil to a boil in a medium saucepan over high heat.
Simmer briskly until liquid thickens and reduces by half, 8-10 minutes.
Off heat, remove ginger using a slotted spoon.
Season broth with salt and pepper to taste and cover to keep warm.
Meanwhile, add 1 tablespoon salt and greens to boiling water; cook until greens are almost tender, 2-4 minutes.
Add in noodles, stir to separate, and cook until both greens and noodles are tender, about 2 minutes longer.
Reserving ¼ cup noodle cooking water, drain noodles and greens, and return to pot.
Add sauce and reserved cooking water to pot; cook over med-low heat, stirring to meld flavors, about 1 minute.
Adjust seasonings with salt and pepper; serve immediately.
**If you cannot find fresh udon noodles, then use dry; add them to the pot of boiling water in step 6 first; cook for about 3minutes, then add the greens and continue cooking until both the noodles and greens are tender, another 2-3 minutes.