In a large mixing bowl, combine the shredded chicken, cream cheese, optional sausage, sliced green onions, cheddar cheese, hot sauce, and Cajun seasoning. Mix until well incorporated, about 5 minutes.

Lay an egg roll wrapper on a clean surface in a diamond shape. Spoon about 2 tablespoons of the filling at the bottom corner of the wrapper. Fold the bottom corner over the filling, tuck in the sides tightly, and roll it up towards the top corner. Seal with beaten egg.

In a deep skillet, pour vegetable oil to a depth of about 2 inches and heat to 350°F. Test with a small wrapper piece; if it sizzles, it's ready.

Carefully add the egg rolls to the hot oil in batches. Fry for 3–4 minutes until golden-brown. Use a slotted spoon to remove and drain on paper towels.

Arrange crispy egg rolls on a serving platter and serve hot with dipping sauces.