Heat oil in a large heavy frypan over medium high heat. Season chicken breasts with salt and pepper. Cook until lightly browned, about 3 minutes on each side. Cover and reduce heat to low. Cook until no longer pink and juices run clear, about 10 minutes. Remove chicken from pan and keep warm.
Drain excess oil from pan. Stir in lime juice and cook over low heat until it begins to bubble. Stir in butter 1 tablespoon at a time. Continue stirring until butter becomes opaque and sauce thickens. Remove from heat and stir in dill weed. Spoon sauce over chicken and serve.