Make the Wonton Filling: In a medium bowl, combine the ground chicken, chopped green onions, 1 tablespoon soy sauce, 1 teaspoon grated ginger, 1 teaspoon sesame oil, and 1/2 teaspoon white pepper. Mix well until combined.

Fill the Wontons: Place a wonton wrapper on a flat surface. Place about a teaspoon of filling in the center.

Seal the Wontons: Moisten the edges of the wrapper with water. Fold in half diagonally to form a triangle, pressing out air and sealing edges. Optionally, bring the two bottom corners of the triangle together and pinch to create a nurse's cap shape.

Repeat: Continue filling and sealing wontons until all filling is used.

Cook the Wontons: Bring a large pot of water to a boil. Gently drop wontons into boiling water in batches. Cook for 3-5 minutes, until they float and filling is cooked. Remove with a slotted spoon and set aside.

Prepare the Broth: In a separate large pot, bring 8 cups chicken broth, 1 tablespoon soy sauce, 1 teaspoon sesame oil, 1/4 teaspoon white pepper, and optional ginger slices to a simmer. Adjust seasoning to taste.

Assemble the Soup: Divide cooked wontons among serving bowls. Ladle hot broth over wontons.

Garnish and Serve: Top with chopped green onions and a few drops of sesame oil. Add blanched greens if desired. Serve immediately.