Heat 1 Tbsp. of vegetable oil in a large nonstick pan over medium heat. Saute the ginger and garlic until they are softened and sizzling, about 1 minutes.
Add the carrots, scallions and cabbage to the pan. Saute until softened.
Stir in the ground chicken, soy sauce, sugar and sesame oil. Break up the chicken into small pieces using a wooden spoon or potato masher. Once the chicken is cooked through, remove the pan from the heat.
Assemble the egg rolls: Place one egg roll wrapper on a flat surface in a diamond shape (so that one point is facing you). Moisten the top corner with water. Place 1/2 cup of filling in the bottom quarter of the diamond. Roll the wrapper up over the filling, tucking the filling in. Fold the right and left corners over the filling and finish rolling the egg roll. Seal the egg roll with the moistened corner. Set aside, seam side down. Repeat the process with 7 more egg rolls.
Heat 1/2 inch of vegetable oil over medium-high heat in a deep skillet. Fry the egg rolls until they are a deep golden brown color. Remove with a slotted spoon onto a wire rack. Blot with paper towels to absorb any excess oil.