Bring the broth to a simmer in a medium pot. Stir in soy sauce, rice vinegar, ginger, and garlic.
Add mushrooms and carrots. Simmer until slightly tender, about 5 minutes.
Slide in the frozen dumplings. Gently simmer until cooked through, 5 to 7 minutes.
Stir in bok choy and white pepper. Cook 2 minutes until greens are tender-crisp.
Finish with sesame oil and most green onions. Taste and adjust seasoning.
Ladle into bowls. Top with remaining green onions, cilantro, and red pepper flakes if using.