In a bowl, combine shredded chicken, soy sauce, sesame oil, garlic, ginger, green onions, bell peppers, and carrots. Mix well.
Place a small spoonful of the mixture onto each wonton wrapper.
Brush the edges of the wrappers with the beaten egg and fold to seal.
Heat vegetable oil in a pan over medium heat.
Fry the wonton bites until golden brown and crispy.
Serve hot with dipping sauce of choice.