In a large mixing bowl, combine the ground chicken, shredded cabbage, chopped onion, sliced green onions, salt, sugar, soy sauce, ground white pepper, minced garlic, and sesame oil (if using). Mix well until all the ingredients are evenly incorporated.
Take a wonton wrapper and place it on a clean surface. Spoon about 1 tablespoon of filling into the center of the wrapper. Dip your finger in the bowl of water and moisten the edges of the wrapper. Fold the wrapper in half to create a triangle, pressing out any air as you seal the edges. Make sure the wonton is tightly sealed to prevent any filling from escaping during frying. Repeat this process until all the filling is used.
In a large frying pan or deep fryer, heat about 2 inches of peanut oil over medium-high heat. To test if the oil is hot enough, drop a small piece of wonton wrapper into the oil. If it sizzles and rises to the surface, the oil is ready.
Carefully drop the sealed wontons into the hot oil in batches, being sure not to overcrowd the pan. Fry them for about 3 to 4 minutes, or until they are golden brown and crispy. Use a slotted spoon to remove the wontons from the oil and drain them on a plate lined with paper towels.
Once all the wontons are fried, serve them hot with your favorite dipping sauce, such as sweet and sour sauce or soy sauce. Enjoy your homemade Chicken Fried Wontons with family and friends!