Bring broth, ginger, and garlic to a gentle simmer in a large pot; simmer 8 minutes to infuse flavor.
Remove ginger and garlic pieces, then whisk miso into a ladle of hot broth until smooth before returning it to the pot.
Add oyster mushrooms and simmer 4–5 minutes until tender. Stir in soy sauce and sesame oil.
Cook udon separately according to package, drain, and divide between bowls.
Add greens to the broth to wilt, taste and adjust seasoning, then ladle over noodles. Garnish with scallions, sesame seeds, and chili flakes.