In a large bowl, combine the shredded chicken, buffalo wing sauce, cream cheese, cheddar cheese, mozzarella cheese, and green onions until well mixed.

Lay one egg roll wrapper on a clean surface in a diamond shape. Place about 2 to 3 tablespoons of the filling near the center of the wrapper.

Fold the bottom corner over the filling, then fold in the left and right corners. Brush the top corner with beaten egg and roll tightly to seal. Repeat with the remaining wrappers and filling.

Heat about 2 inches of vegetable oil in a deep skillet or pot to 350°F (175°C).

Carefully fry the egg rolls in batches for 2 to 3 minutes per side, or until golden brown and crisp.

Transfer the egg rolls to a paper towel-lined plate to drain briefly.

Serve warm with ranch or blue cheese dressing for dipping.