In a wok or large skillet, stir-fry kimchi, bean sprouts and carrots in 1-1/2 teaspoons oil until tender; set aside.
In a small skillet, cook beef over medium heat until no longer pink; drain. Add to the vegetable mixture. Stir in the onions, sesame seeds, ginger, garlic, sesame oil, salt and pepper.
Place about 1 tablespoon of filling in the center of each wonton wrapper. Combine egg and water. Moisten wonton edges with egg mixture; fold opposite corners over filling and press to seal. At this point you can freeze them.
Heat remaining vegetable oil in a large skillet. Cook wontons in batches for 1-2 minutes on each side or until golden brown, adding additional oil if needed.