In a mixing bowl, combine the ground chicken or shrimp, green onions, ginger, garlic, soy sauce, sesame oil, salt, and pepper. Mix well to form the filling for the wontons.
Place a wonton wrapper on a clean surface. Spoon about 1 teaspoon of the filling into the center of the wrapper.
Wet the edges of the wrapper with a little water, fold it in half, and press the edges together to seal. For a more decorative shape, pinch the corners together. Repeat with the remaining wrappers and filling.
In a large pot, bring the chicken broth to a gentle boil over medium heat.
Add the sliced mushrooms and bok choy to the broth. Cook for about 5 minutes until the vegetables are tender.
Carefully drop the wontons into the pot. Allow them to cook for about 5-7 minutes, or until they float to the surface and are cooked through.
Taste the broth and adjust the seasoning with more salt if needed.
Serve the wonton soup hot, garnished with fresh cilantro on top.