In a heavy non-reactive casserole, warm the olive oil over low heat. Slowly cook the onion and carrot until they began to turn golden, then arrange the artichokes in the casserole in a single layer. Add the bay leaf, thyme, the 2 whole cloves of garlic, and salt and pepper to taste. Pour the wine into the casserole; add water until the artichokes are just covered. Cook over medium heat for 15 minutes, covered, then remove the lid, add the asparagus and reduce the cooking liquid over high heat to thicken the sauce. While waiting for the reduction, saute the snails in butter. Mince the remaining garlic cloves and mix in the basil and parsley. Off the heat, add the minced garlic mixture and snails, check for seasoning. Stir well and serve immediately. Peel the artichoke bottoms & stems, and rub each with half a lemon as you finish trimming it. Place them in cold water acidulated with the juice from the remaining lemon half. Spread the leaves of each artichoke, and use a small spoon to remove the choke; return to the lemon water.
Preparing the Artichokes: cut the stems to about 1 1/2 inches, cut about 3/8-inch off the artichoke tops, then break off 2 or 3 leaves from the base.