Bring the chicken broth to a simmer in a medium pot.
Stir in ginger, garlic, soy sauce, and white pepper. Simmer 3 minutes to infuse.
Add mushrooms and cook until just tender, about 3 minutes.
Slide in the wontons and cook gently until they float and are heated through.
Stir in bok choy and cook until crisp-tender, 2 minutes. Adjust salt if needed.
Finish with sesame oil and green onions. Ladle into bowls and drizzle chili oil.