Season chicken with salt and pepper. Heat one tablespoon of oil and one tablespoon of butter in a skillet over medium-high heat. Fry breasts on both sides until golden browned and completely cooked through (about five to six minutes per side). Once cooked, transfer to a plate and tent with foil to keep warm.

Heat remaining butter and oil in the pan. Fry onion and garlic until garlic is fragrant (about two minutes). Add olives and tomatoes and cook for two minutes, or until the tmatoes just begin to soften. Turn off the heat and stir through basil. Add the chicken back into the pan, and spoon the pan juices and tomato/garlic mixture all over the chicken.

Serve with pasta and a salad.