In a large bowl, mix ground chicken, carrots, cabbage, mushrooms, garlic, and green onions.
Add soy sauce, fish sauce, sesame oil, pepper, and sugar to the mixture, stirring until combined.
Soak rice paper wrappers in warm water for 3-5 seconds and lay flat.
Place 2 tablespoons of the filling on the lower third of the wrapper.
Fold the bottom over the filling, fold in the sides, and roll tightly.
Heat oil in a deep pan to 350°F (175°C).
Fry the egg rolls in batches for 5-7 minutes until golden brown, turning occasionally.
Remove the egg rolls and drain excess oil on paper towels or a wire rack.
Serve with Nuoc Cham dipping sauce, lettuce leaves, and herbs.