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 & chicken broth, and stir until melted. Season with chives and dill. Spoon sauce over chicken, and serve immediately.