Heat a large skillet over medium-high heat. Cook and stir beef and pork in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and discard grease. Transfer meat mixture to a large bowl.

Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add cabbage, cover, and steam until tender, 2 to 4 minutes.

Heat oil in a deep-fryer or large saucepan to 360 degrees F (182 degrees C).

Mix cabbage, bean sprouts, celery, green onion, soy sauce, cornstarch, sesame oil, salt, and pepper into meat mixture. Spoon mixture into the center of each wonton wrapper. Brush beaten egg onto two edges of each wrapper and fold wrapper around filling, sealing the edges together.