Make butter: Put ingredients into bowl of a mixer and beat until smooth.
Prepare lamb: Preheat oven to 425°. Heat a large cast-iron frying pan over high heat until very hot. Add oil and swirl pan to coat. Put lamb in center of pan. Put garlic halves on either side of lamb, cut sides up, and lay mint over garlic. Cook lamb, turning once, until well browned, about 4 minutes on each side.
Put pan with lamb in oven to roast until meat reaches 125° on a meat thermometer, 12 to 14 minutes for medium-rare.
Transfer lamb to a cutting board and tent with foil. Let rest 10 minutes before slicing. Serve with mint and garlic on the side and a dollop of anchovy butter on the warm lamb slices.