Combine first 6 ingredients in a small bowl; stir until blended. Shape butter mixture into a 3-inch stick; cover and freeze 45 minutes or until firm.
Place chicken between two sheets of heavy-duty plastic wrap; flatten to 1/4-inch thickness, using a meat mallet or rolling pin. Season chicken with 1/4 teaspoon salt and 1/8 teaspoon pepper.
Cut butter stick into 6 pats; place 1 pat in center of each chicken breast half; fold long sides of chicken over butter, fold ends over, and secure with wooden picks.
Combine egg and water, stirring well. Dredge each chicken roll in flour, dip in egg mixture, and dredge in breadcrumbs. Cover and chill rolls 1 to 8 hours.
Heat 1/2 cup oil to 350° in a large skillet; fry chicken in hot oil 3 minutes or until browned, turning to brown rolls evenly. Transfer chicken to a baking sheet; bake at 350° for 16 to 18 minutes.