Preheat oven to 450 degrees, with rack in lower third. Rub lamb with a generous amount of salt and pepper, then rub with oil. With the tip of a sharp paring knife, cut twenty 1/2-inch-deep slits all over lamb; insert garlic slivers into openings.

Place lamb on a roasting rack set on a rimmed baking sheet; place in oven. Immediately reduce oven temperature to 325 degrees. Roast lamb until an instant-read thermometer inserted in the thickest part (avoiding bone) registers 125 degrees to 135 degrees for rare, or 135 degrees to 140 degrees for medium, 1 1/4 to 1 3/4 hours. Remove from oven; let rest 10 to 15 minutes before carving.

Meanwhile, make the mint sauce: In a small saucepan, bring vinegar, sugar, and water to a boil. Reduce heat; simmer until liquid is syrupy and reduced to 1 cup, 10 to 15 minutes. Remove from heat; stir in mint, and let cool completely.