In a small bowl combine green onion, parsley, and garlic; set aside. Cut chilled butter into four 2-1/2x1/2-inch sticks. In a shallow bowl stir together egg and water. Place flour in another shallow bowl. Place bread crumbs in a third shallow bowl. Set aside.
Place a chicken breast half between 2 pieces of plastic wrap. Using the flat side of a meat mallet pound chicken lightly into a rectangle about 1/8 inch thick. Remove plastic wrap. Sprinkle with salt and pepper. Sprinkle with one-fourth of the green onion mixture. Place a butter piece in center of chicken piece. Fold in sides; roll up. Repeat with remaining chicken.
Coat chicken rolls with flour. Dip in egg mixture; coat with bread crumbs. Dip in egg mixture again; coat with with additional bread crumbs. (Coat ends well to seal in the butter.) Place coated chicken in 2-quart rectangular baking dish. Cover and chill for 1 to 24 hours.
In a large skillet melt the 1 tablespoon butter over medium-high heat; add oil. Add chilled chicken rolls. Cook about 5 minutes or until golden brown, turning to brown all sides. Place rolls in a 2-quart rectangular baking dish. Bake, uncovered, in a 400 degree F oven for 15 to 18 minutes or until chicken is no longer pink (170 degrees F). Spoon any drippings over rolls.