Place breast halves, one at a time, between 2 sheets of plastic wrap. With flat side of a mallet or with a small frying pan, gently pound breasts until each is about 1/4" (6mm) thick; set aside.
In a pie pan, combine bread crumbs, Parmesan cheese, 1 teaspoon of the oregano, the sale, and peper; set aside. In a small boul, mash butter with parsley and remaining 1/2 teaspoon oregano.
Spread about 1/2 teaspoon of the herb-butter mixture on each breast about 1 inch (2.5cm) from lower edge; lay a piece of jack cheese on butter mixture. Fold lower edge of breast over filling, then fold in sides and roll up to enclose filling.
roll each bundle in egg mixture and drain briefly; then roll in bread crumb mixture until evenly coated. Place bundles, seam side down, without touching, in a 9x15" baking pan. Cover and refridgerate for at least 4 hours, or overnight.
Bake, uncovered, in a 425F oven until chicken is no longer pink in center; cut to test (about 30 minutes).