Coat chicken breasts with egg, and dip in bread crumbs. Place on a wire rack, and allow to dry for about 10 minutes.
Heat olive oil in a large skillet over medium heat. Place chicken into the skillet, and fry for 3 to 5 minutes on each side. Remove to a platter, and keep warm.
Drain grease from the skillet, and squeeze in lime juice. Cook over low heat until it boils. Add butter, and stir until melted. Season with chives and dill. Spoon sauce over chicken, and serve immediately.