Combine ground pork, soy sauce, sesame oil, ginger, green onions, cornstarch, and white pepper in a bowl. Mix until uniform and slightly sticky. Cover and let rest for 10 minutes.
Place 1 teaspoon of filling in the center of each wrapper. Dampen edges with water, fold diagonally to form a triangle, and press to seal. Bring the two bottom corners together and press to form the classic wonton shape. Place on a floured baking sheet and cover with a damp towel.
Combine chicken broth, soy sauce, sesame oil, white pepper, ginger, and garlic in a large pot. Bring to a gentle simmer and adjust seasoning to taste.
Carefully lower wontons into the simmering broth one at a time. Cook for 4-5 minutes until they float to the top and wrappers look slightly translucent. Remove cooked wontons with a slotted spoon if cooking in batches.
Whisk eggs in a small bowl. In a separate cup, stir cornstarch and cold water until dissolved. Pour the slurry into the broth and stir gently for 30 seconds. Reduce heat to medium-low.
Stir the broth in a slow, steady circular motion while slowly pouring in the beaten eggs in a thin stream. Let the eggs set for about 30 seconds to form delicate ribbons.
Gently return all cooked wontons to the pot. Ladle the soup into bowls and top with sliced green onions and a drizzle of sesame oil. Serve immediately.