In a large bowl, mix the shredded chicken, buffalo sauce, cream cheese, mozzarella, and cheddar cheese until well combined and creamy.
Lay one egg roll wrapper on a clean surface in a diamond shape. Place 2–3 tablespoons of filling in the center. Fold the bottom corner over the filling, then fold in the sides. Brush the top edge with beaten egg and roll tightly to seal.
In a deep skillet or fryer, heat about 2 inches of oil to 350°F (175°C). Prepare a paper towel-lined plate for draining.
Carefully place a few egg rolls at a time into the hot oil. Fry for 3–4 minutes, turning occasionally, until golden brown and crispy. Remove with tongs and drain on the paper towel-lined plate.
Serve hot with ranch or blue cheese dressing on the side. Sprinkle with green onions if desired.