In a pan over low heat, slowly simmer garlic in six ounces of olive oil until tender, to impart the taste of the garlic in the oil. This will take about one hour. Let cool, and remove garlic from the oil. In a small bowl, slowly whip the garlic infused oil and the egg yolk. Add salt and pepper to taste, set aside. This will create the aioli. Rub loin of lamb with garlic, thyme, salt and pepper and two ounces of oil and let the meat "marinate" for at least I hour, and most overnight. Once the rub has saturated the loin, grill loin in a heavy skillet on high heat to sear all sides. Make sure the loin is well browned on all sides. Transfer to a 350 degree oven for two to three minutes for medium rare, longer for more well done. Let the loin rest for three minutes before slicing it thinly and filling the Brioche bun. Top with one teaspoon of the aioli and wrap well for the picnic basket.
Recipe Courtesy of John Villa, Exec. Chef, Park View at the Boathouse, New York, NY