Heat a large wok or skillet over medium-high heat and add the ground chicken, minced ginger, and minced garlic.
Cook, breaking up the meat, until the chicken is crumbly and no longer pink; drain any excess fat and return the pan to the heat.
Add the shredded cabbage and carrot mix, sliced green onions, salt, sugar, sesame oil, and Chinese five-spice powder to the chicken.
Stir and cook until the cabbage and carrots are tender, about 3–5 minutes, then remove from heat and let the filling cool slightly so it is easy to handle.
Lay one egg roll wrapper on a clean surface with a corner pointing toward you. Place about 2 tablespoons of the chicken mixture near the corner closest to you.
Brush the edges of the wrapper with beaten egg, fold the corner over the filling, tuck in the sides, and roll tightly to enclose the filling; repeat with remaining wrappers and filling.
Heat a heavy-bottomed pot of vegetable oil to 375°F (190°C) or heat a deep fryer to the same temperature; there should be about 2 inches of oil in the pot.
Fry the egg rolls in batches until golden brown and crispy, about 3–4 minutes per batch, turning as needed for even color.
Use a slotted spoon or tongs to transfer the fried egg rolls to a paper towel-lined plate to drain before serving.