Place chicken breast halves between plastic (like a Walmart bag). Pound slightly with mallet. Sprinkle with salt and pepper.
Heat 1 tbsp each of oil and butter in large skillet.
Cook chicken over high heat for 4 minutes on each side. (Don't cook longer or they'll be overcooked and dry. Juices should run clear.)
Transfer to warm serving platter.
Add chives or green onion, lime juice, parsley, and mustard to pan. Cook 15 seconds, stirring constantly.
Whisk in broth, stir until sauce is smooth. Whisk in remaining butter and oil.
Pour sauce over chicken. Serve immediately.
Good with noodles with tomato sauce, steamed broccoli, and fresh salad.