Heat 1 tablespoon vegetable oil in a large skillet over medium heat.
Add minced garlic and chopped onion to the skillet and sauté until fragrant and translucent, about 2 minutes.
Add ground chicken to the skillet and cook until no longer pink, breaking it apart with a spatula, about 5 minutes.
Stir in julienned carrot, shredded cabbage, and bean sprouts. Cook for 3 minutes until vegetables are slightly softened.
Add soy sauce, oyster sauce, and ground black pepper to the mixture. Stir well to combine and cook for another 2 minutes. Remove from heat and let the filling cool slightly.
Place one spring roll wrapper on a clean, flat surface with one corner pointing toward you (diamond shape).
Spoon about 2 tablespoons of the chicken filling near the corner closest to you.
Fold the corner over the filling, then fold in the left and right corners toward the center.
Roll the wrapper tightly toward the far corner, sealing the edge with a brush of beaten egg.
Repeat the rolling process with the remaining wrappers and filling.
Heat vegetable oil in a deep frying pan or wok to 350°F (175°C).
Fry spring rolls in batches, turning occasionally, until they are golden brown and crispy, about 3 to 4 minutes per batch.
Remove spring rolls with a slotted spoon and drain on paper towels.
Serve hot with your favorite dipping sauce.