To make the shrimp filling, in a food processor, combine half of the shrimp with salt, white pepper, and oyster sauce. Blend until smooth and sticky. Add the remaining shrimp and pulse briefly to keep some texture. Stir in the ginger and scallions.
To wrap the wontons, wet the edges of a wonton wrapper with water. Place about 1 teaspoon of filling in the center. Fold into a triangle, bring the corners together, and press to seal. Repeat with the remaining wrappers.
To make the broth, in a pot over high heat, bring the chicken stock to a boil. Season with salt, soy sauce, sesame oil, and white pepper.
To cook the bok choy and wontons, in another pot over high heat, bring water to a boil. Add the bok choy and blanch for about 30 seconds, then remove. Return the water to a boil and add the wontons. Cook for about 2 minutes. Pour in 1 cup of room-temperature water and continue cooking another 2 minutes until the wontons are translucent and cooked through.
To serve, place the wontons in serving bowls. Ladle hot broth over them and add the bok choy. Serve immediately.