1. Rinse lamb and pat dry. Chop 10 cloves of garlic. In an 8- by 10-inch baking dish, mix half the chopped garlic, 1/2 cup olive oil, vinegar, shallots, thyme, parsley, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Add lamb and turn to coat. Cover and chill at least 4 hours or up to 1 day, turning occasionally.
2. On a 12- by 12-inch sheet of foil, mix remaining 20 cloves garlic with 1/2 tablespoon olive oil to coat; seal foil around garlic. Bake in a 350° regular or convection oven until cloves are soft when pressed, about 45 minutes, turning packet over halfway through baking.
3. While garlic is baking, in a 10- to 12-inch frying pan over high heat, boil wine until reduced to 1 cup, about 10 minutes. Add broth and boil, stirring occasionally, until mixture is reduced to about 1 1/2 cups, about 20 minutes. Whisk in 1 tablespoon butter, then stir in mint and the roasted garlic. Keep warm over low heat.
4. In a 5- to 6-quart pan, combine potatoes and remaining chopped garlic; add water to cover. Bring to a boil over high heat; cover, reduce heat, and simmer until potatoes mash easily when pressed, 20 to 25 minutes.
5. Meanwhile, heat cream with the remaining 1/2 cup butter in a microwave-safe container in a microwave oven at full power (100%), stirring at 20-second intervals, until butter is melted and mixture is steaming (do not boil), about 1 1/2 minutes total.
6. Drain potatoes and garlic. Mash with a potato masher or a mixer until almost smooth. Add cream mixture and mash to desired consistency; if potatoes are thicker than desired, add a little milk. Add salt and pepper to taste. Keep warm over low heat if necessary.
7. While potatoes cook, lift lamb from marinade and drain well (discard marinade). Lay on a barbecue grill over a solid bed of medium-hot coals or medium-high heat on a gas grill (you can hold your hand at grill level only 3 to 4 seconds); close lid on gas grill. Turn occasionally until lamb is browned on all sides and a thermometer inserted in center of thickest part reaches 135° for medium-rare (pink in the center), 20 to 25 minutes, or until done to your liking. Transfer lamb to a board and let rest in a warm place at least 5 minutes.
8. Mound mashed potatoes equally on plates. Cut and remove string from lamb; slice loin crosswise and fan slices over potatoes. Spoon sauce with garlic around meat and mashed potatoes.
Notes: Have the butcher bone a lamb loin (1 3/4 to 2 lb.; make sure you don't get a tenderloin or the sirloin) and roll and tie it into a roast about 8 inches long and 2 1/2 inches wide. You can marinate the lamb and prepare the sauce through step 3 up to 1 day ahead; bring sauce to a simmer over low heat before serving. Pour Buttonwood Farm Winery's "Trevin" Bordeaux blend to match the Cabernet sauce here.