Pre-soak the dried shiitake mushrooms in water for 20 minutes. If using fresh mushrooms, skip this step. Squeeze out any excess water from the dried mushrooms and remove and discard the stems. Dice the mushrooms into small cubes.
Combine the diced mushrooms, ground chicken, and chopped scallions in a bowl. Season with Chinese Shaoxing cooking wine, soy sauce, oyster sauce, salt, sugar, sesame oil, and ground white pepper. Stir well with a spatula or a pair of chopsticks, and marinate the filling in the refrigerator for 30 minutes.
Prepare a small bowl of water nearby. Place one wonton wrapper in your palm and add one teaspoon of filling to the center. Wet your finger and dab the edges along one side of the wrapper with water. Fold the wonton wrapper in half to form a rectangle, then pinch the edges to seal the filling inside.
Dab one bottom corner of the wonton wrapper with water. Gather the two corners of the wrapper using your thumb and index fingers on both hands, and press down firmly. Ensure the wontons are sealed tightly. Repeat until all the filling is used.
Prepare the soup by adding chicken bouillon powder, salt, white pepper, and sesame oil to a bowl. Pour boiling water into the bowl and stir well. In a separate pot, bring water to a boil. Drop the wontons into the boiling water and stir gently to prevent them from sticking together. Cook the wontons uncovered until they float to the top and are fully cooked.
Turn off the heat. Drain the wontons using a strainer and transfer them into the soup. Serve immediately.