Mash green onion and ginger in a bowl with water. Mix in pork, 2 tablespoons rice wine, soy sauce, 2 teaspoons salt, sesame oil, and white pepper.
Combine shrimp, 1 tablespoon rice wine, 1 teaspoon salt, mushrooms, and celery in a bowl. Add to the pork mixture; stir until filling is well combined.
Separate and place wonton wrappers onto your work surface. Spoon about 1 tablespoon of the pork-shrimp filling onto the center of each wrapper. Use your finger or a pastry brush to lightly moisten edges of wonton wrappers with water. Fold one corner of the wrapper over the filling onto the opposite corner to form a triangle. Press edges together to seal.
Bring the chicken stock to a rolling boil. Add wontons and cook until they float to the top, about 5 minutes.