In a small bowl, combine olive oil, parsley, rosemary, and garlic. Season with salt, to taste. Set aside.
Set a 12-inch oven-safe saute pan over medium-high heat. Season the lamb with 1 tablespoon of salt and 1 1/2 teaspoons of the black pepper. Add 1 1/2 teaspoons of the oil to the saute pan and place the lamb, fat side down, in the pan. Sear the lamb until the fat has rendered and the skin is golden brown, about 3 minutes. Continue to cook the lamb, turning to cook evenly on each side, about 2 minutes each side. Baste them with the herb and oil mixture. Transfer the pan to the oven and cook until the lamb is rare, about 5 minutes. Alternatively, you can grill the lamb chops.
Make the balsamic drizzle while the lamb is cooking: Set a 1-quart saucepan over medium heat and add the remaining 1 1/2 teaspoons of olive oil. Once the oil is hot, add the shallots to the pan and sweat until translucent, about 1 minute. Add the garlic to the pan and sweat for 30 seconds. Pour the balsamic vinegar into the pan and bring to a boil. Add the rosemary and allow the balsamic to gently boil and reduce until only about 1/4 cup of balsamic remains, about 10 minutes. Season with 1 teaspoon of salt and 1/2 teaspoon of black pepper. Remove from the heat and swirl the butter into the pan. Remove the rosemary sprigs before using.
When the lamb is rare, remove from the oven and let rest for 5 minutes before serving. To serve, slice each double chop in half, and drizzle with the balsamic reduction.