Mix all the ingredients of the rub. Cut the lamb racks into 2 half-racks and season the lamb generously with the rub. Set on a plate and let rest for 25 minutes at room temperature.
Heat oil in a large skillet over medium-high heat. Lay the lamb racks in the pan, fat side down. Brown the lamb turning occasionally with thongs until all sides are golden brown. Reduce the heat to medium and add butter, 3 springs of rosemary, and garlic to the skillet. Continue cooking basting the lamb with butter and turning occasionally until lamb internal temperature of the lamb is 130°F for medium. Remove lamb from heat and allow resting for 5 minutes. Cut the lamb chops and serve. Garnish the plate with fresh rosemary springs.
Lamb is traditionally paired with mint jelly.
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