Dice chicken into small cubes. Transfer to a mixing bowl. Add celery, blue cheese, cream cheese, and hot sauce. Season with salt, black pepper, and cayenne pepper. Mix with a spoon until filling is well combined.

Beat egg and water together. Brush egg wash over the edges of 1 wonton wrapper. Place a heaping tablespoon of filling near one edge. Spread out with your fingers, leaving a 1/2-inch border. Roll edge up over filling. Crimp in both edges and roll over once more. Fold in about 1/4 inch of dough on either side to seal the ends. Apply more egg wash on top and continue rolling. Place on a plastic-lined baking pan. Repeat with remaining wonton wrappers and filling.

Fill a skillet with enough oil to reach halfway up the egg rolls. Heat to 350 degrees F (175 degrees C) in a skillet over medium heat. Add a batch of egg rolls; fry until browned and crispy, about 2 minutes per side. Drain on a paper towel-lined plate. Fry remaining egg rolls.