Place each chicken breast half between two pieces of heavy plastic wrap. Working from center to edges, pound lightly with the flat side of a meat mallet to 1/8-inch thickness. Remove plastic wrap. Season lightly with salt and pepper.
To assemble, place 1 piece of the cheese in the center of each chicken piece. Sprinkle each piece with half of the green onion and half of the finely chopped nuts. Fold in the sides of each piece; roll up into a spiral, pressing the edges to seal.
In a shallow dish, beat together egg and the water. Place flour in another shallow dish; place ground nuts in a third shallow dish. Coat each chicken roll with flour, dip into egg mixture, and finally roll in nuts to coat. Transfer to a 9-inch pie plate. Cover and chill for at least 30 minutes or up to 24 hours.
In a medium skillet, cook the chicken rolls in hot butter over medium heat about 5 minutes or until golden, turning to brown all sides. Return to pie plate. Bake in a preheated 400-degree oven for 15-18 minutes or until no longer pink in the center and an instant-read thermometer inserted into the center registers 170-degrees. Transfer to serving plates.