In a large bowl, combine shredded chicken, shredded carrots, shredded cabbage, chopped green onions, minced garlic, soy sauce, sesame oil, and ground black pepper. Mix well until all ingredients are evenly incorporated.
Place one spring roll wrapper on a clean flat surface with one corner pointing towards you (diamond shape).
Spoon about 2 tablespoons of the chicken mixture near the corner closest to you.
Fold the corner over the filling, then fold in the left and right corners towards the center.
Roll the wrapper tightly away from you, sealing the top edge with a little of the beaten egg to secure the spring roll.
Repeat the wrapping process with the remaining wrappers and filling.
Heat vegetable oil in a deep frying pan or wok over medium-high heat until it reaches 350°F (175°C).
Carefully fry 3-4 spring rolls at a time for 3-4 minutes or until golden brown and crispy, turning occasionally for even cooking.
Remove the spring rolls with a slotted spoon and drain on paper towels.
Serve hot with your favorite dipping sauce.