In a shallow dish, combine the garlic, mustard, 3 tablespoons olive oil, herbs, and ½ teaspoon each of salt and pepper. Add the lamb chops and massage the mixture into both sides of each chop. Cover and marinate in the refrigerator for a minimum of one hour and up to 8 hours. Try to flip the chops halfway to ensure even flavor throughout.
A half hour prior to cooking, let the chops sit out at room temperature for 15 minutes. Preheat the oven to 425℉. Pat the chops dry with a paper towel, then season both sides of each chop with coarse salt and pepper.
Preheat a cast iron skillet over medium high heat. Add the butter and remaining tablespoon of olive oil. Sear the lamb chops for exactly two minutes on each side. Use a spoon to baste the chops with the butter and oil from the pan once you flip them. Transfer the hot skillet to the oven and continue cooking for 4-6 minutes for medium rare or a thermometer inserted into the center of two chops stacked together reads 135℉.
Cover loosely with foil and let the chops rest for 5-10 minutes before serving hot.