Combine ground pork, finely chopped shrimp, minced green onions, grated ginger, soy sauce, sesame oil, rice vinegar, egg white, white pepper, and salt in a medium bowl. Mix until evenly combined, then cover and refrigerate for 10 minutes.
Clear a clean, dry surface and place the wonton wrappers in a stack covered with a damp towel. Beat one egg in a small bowl for sealing, and dust a baking sheet lightly with cornstarch to hold finished wontons.
Place a wonton wrapper diamond-style on your surface, add 1 teaspoon of filling to the center, brush egg wash on all four edges, then fold into a triangle and press firmly to seal. Bring the two bottom corners together and pinch to create the classic cap shape. Repeat with remaining wrappers.
Add chicken broth, soy sauce, sesame oil, grated ginger, white pepper, sugar, and turmeric to a large pot over medium-high heat. Bring to a gentle boil, then reduce to a steady simmer and taste for seasoning.
Gently lower wontons into the simmering broth one at a time and cook for 8 to 10 minutes, until they float and the wrappers look slightly translucent. Remove wontons with a slotted spoon and set aside in serving bowls.
Whisk together 2 tablespoons cornstarch and 3 tablespoons cold water in a small bowl until completely smooth with no lumps. Set aside until ready to use.
Bring the broth back to a full rolling boil, stir in the cornstarch slurry, then slowly pour the beaten eggs in a thin stream while stirring the broth in a gentle circular motion. Remove from heat immediately once all eggs are added.
Place 4 to 6 wontons in each bowl, ladle the hot egg drop broth over them, and garnish with sliced green onions, white pepper, and an optional drizzle of sesame oil. Serve immediately.