Preheat the oven to 410F.
Meanwhile, prepare the chicken breasts. Place the chicken breasts, one at a time, in a gallon zip lock bag and, using a cast iron skillet, give them a good whack to break the bones and slightly flatten them. This will help get a more uniform cooking. Rub the breasts with salt, pepper and minced garlic. Set aside.
Heat the butter and the oil in a large cast iron pan over medium-high heat. As soon as the butter turns dark brown, add the chicken breasts skin side down. Sear for two minutes or until the skin becomes dark golden brown. Flip the breasts and sear for another 2 minutes.
Move the pan into the preheated oven and bake for about 17-20 minutes, until the internal temperature reaches 160F. Remove the breasts from the oven and let them rest for 5 minutes. It’s best to remove the breasts from the pan as hot cast iron will continue cooking the meat for too long. Serve with chopped parsley and lime wedges, and your favorite side dish.