Heat olive oil in a large skillet over medium-high heat.
Add ground chicken and cook, breaking it apart with a spoon, until fully cooked and no longer pink, about 5-7 minutes.
Add minced garlic, shredded cabbage, shredded carrots, and chopped green onions to the skillet. Cook for 3-4 minutes until vegetables are slightly softened.
Stir in soy sauce, grated ginger, and black pepper. Cook for an additional 1-2 minutes, then remove from heat and let the filling cool slightly.
Place one egg roll wrapper on a clean surface with one corner pointing toward you.
Spoon about 1/3 cup of the chicken and vegetable filling onto the center of the wrapper.
Fold the bottom corner over the filling, then fold in the left and right corners toward the center.
Roll the wrapper tightly toward the top corner, sealing the edge with beaten egg to secure the egg roll.
Repeat the rolling process with the remaining wrappers and filling.
Heat a large skillet over medium heat and add 1 tablespoon olive oil.
Place the egg rolls seam side down in the skillet and cook for 3-4 minutes on each side until golden brown and crispy.
Remove egg rolls from the skillet and drain on paper towels before serving.