Preheat your oven to 180 C/350 F. Season the lamb with salt and pepper on both sides. Add the oil to a cast-iron skillet and place it over medium-high heat.
Add the lamb to the skillet, flesh side down, and let it cook for 2-3 minutes until caramelized. Turn over and sear the other side for another 2-3 minutes. Transfer the skillet to the oven.
Roast the lamb for 12 minutes for medium-rare and 15 minutes for medium. Remove the lamb from the pan and wrap it in aluminum foil. Let it rest for 10 minutes.
While the lamb is resting, prepare the sauce. Return the skillet to medium heat and add the butter. Finely dice the shallot and add it to the pan. Cook it for 2-3 minutes until softened, stirring frequently.
Mince the garlic and add it to the skillet. Cook for 1 minute, then add the broth, mustard, lemon juice, and capers. Stir and bring the sauce to a simmer.
Drop the heat to low and simmer the sauce for 10 minutes. Pour the sauce into a serving bowl. Carve the lamb rack into chops and serve.