In a medium mixing bowl, combine the ground pork or chicken, chopped shrimp, minced garlic, soy sauce, oyster sauce, sesame oil, grated ginger, chopped green onion, and ground white pepper. Mix thoroughly until well combined.
To assemble the wontons, place a wonton wrapper on a clean, dry surface. Spoon about 1 teaspoon of the filling into the center of the wrapper. Dip your finger in water and moisten the edges of the wrapper. Fold the wrapper over the filling to form a triangle, then press to seal the edges, ensuring there are no air pockets.
Bring a large pot of water to a gentle boil. Carefully drop the wontons into the boiling water and cook for about 2-3 minutes or until they float to the surface. Remove them with a slotted spoon and set aside.
In the same pot, bring the chicken broth to a boil. Add the chopped bok choy or Chinese cabbage, soy sauce, oyster sauce, sesame oil, and ground white pepper. Let it simmer for 3-4 minutes until the greens are tender.
Add the cooked wontons to the soup and simmer for an additional 1-2 minutes to allow the flavors to meld.
Ladle the Homemade Wonton Soup into soup bowls, garnish with thinly sliced green onions, and serve hot.