Preheat oven to 475 degrees. In the bowl of a food processor, combine shallot, garlic, mustard, 3 tablespoons olive oil, and mint; season with salt and pepper. Pulse until a medium-textured paste forms, about 30 seconds; set aside.
Heat a cast-iron skillet large enough to accommodate the half-rack of ribs, over medium-high heat. Add the remaining tablespoon olive oil. Season lamb with salt and pepper, and place in the skillet, fat side down. Sear meat on both sides until well browned, 4 to 5 minutes. Remove the skillet from the heat, and transfer the lamb to a clean surface.
Spread the reserved mint paste on the meaty side of the lamb, and place the lamb in a shallow baking pan. Transfer to the oven, and roast until the lamb is medium rare, or until the internal temperature is 130 degrees. 18 to 22 minutes. Remove from the oven, and place lamb on a cutting board; let rest for 5 to 10 minutes before slicing. Slice between the bones, and serve.
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