Chicken Schnitzel

2 1/4 inch thick chicken cutlets (1 1/4 to 1 1/2 pounds total)
1/4 cup all-purpose flour
1/2 teaspoon salt
1/8 teaspoon black pepper
1 large egg
1/2 cup fine dry bread crumbs
2-3 tablespoons olive oil
1 1/2 tablespoons unsalted butter
1 tablespoon chopped fresh flat-leaf parsley

Gently pound each cutlet between 2 sheets of plastic wrap to 1/8-inch thickness with flat side of a meat pounder or with a rolling pin. Stir together flour, salt, and pepper in a shallow bowl or a pie plate. Whisk together eggs and a pinch of salt in another shallow bowl, then put bread crumbs in a third shallow bowl. Pat chicken dry and dredge in seasoned flour, shaking off excess. Dip in eggs, letting excess drip off, then dredge in bread crumbs, coating completely. Heat 2 tablespoons oil in a 12-inch heavy, nonstick skillet over moderately high heat until hot but not smoking. Fry cutlets for about 3 minutes, turning over once, until golden brown and just cooked through to an internal temperature of 165°F. Remove from heat and set aside. Add butter to skillet and heat over moderate heat, swirling skillet, until butter is melted. Stir in parsley and pour sauce over cutlets.