In a medium bowl, combine the softened cream cheese, chopped green onions, soy sauce, garlic powder, and white pepper. Mix thoroughly until all ingredients are fully incorporated and the texture is smooth and consistent.
Create an assembly line by arranging your wonton wrappers, cream cheese filling, beaten egg (in a small bowl), and a baking sheet lined with parchment paper. Keep a damp paper towel over your wonton wrappers to prevent them from drying out while you work.
Place a wonton wrapper on a flat surface with one corner pointing toward you (diamond orientation). Add approximately 1 teaspoon of filling to the center of each wrapper. Using your finger or a small pastry brush, apply beaten egg along the edges of the wrapper.
Fold the wrapper diagonally to form a triangle, pressing firmly to seal the edges and remove any air pockets, which can cause wontons to burst during frying.
For the classic restaurant-style shape, take the two opposite corners of your triangle and bring them together, overlapping slightly. Secure with a dab of egg wash. This creates the characteristic "nurse's cap" shape.
Heat vegetable oil in a large, heavy-bottomed pan to 350°F (175°C). If you don't have a thermometer, test the oil by dropping in a small piece of wonton wrapper – it should bubble and rise to the surface immediately.
Fry the wontons in batches of 6-8 (depending on your pan size) for 1-2 minutes per side until golden brown and crispy. Avoid overcrowding, which can reduce the oil temperature and result in soggy wontons.
Using a slotted spoon, transfer the fried wontons to a plate lined with paper towels to absorb excess oil. For maximum crispiness, arrange them in a single layer rather than stacking.
Serve immediately while hot and crispy, with sweet chili sauce or your preferred dipping sauce on the side.