Heat 1 tablespoon olive oil in a large skillet over medium heat.
Add 1 pound ground chicken, 2 minced garlic cloves, and 1 tablespoon minced fresh ginger; cook, breaking up the meat, until the chicken is browned and cooked through.
Stir in 2 cups shredded cabbage, 1 cup shredded carrot, and 1/4 cup sliced green onions; cook until the vegetables are tender, about 3–5 minutes.
Season the filling with 1/4 teaspoon salt, 1/2 teaspoon sugar, 1 tablespoon sesame oil, and 1/2 teaspoon Chinese five-spice powder; mix well and remove from heat to cool slightly.
Place an egg roll wrapper on a clean surface and put about 2 tablespoons of the filling near one corner.
Fold the corner over the filling, fold in the sides, and roll tightly toward the opposite corner; seal the edge with the beaten large egg.
Heat 3 cups vegetable oil in a deep skillet or pot to 375°F (190°C) using a thermometer.
Fry the egg rolls in batches until golden brown, about 2–3 minutes per side, turning as needed; do not overcrowd the pan.
Transfer fried egg rolls to a paper towel-lined plate to drain briefly before serving.
