Bring a large pot of water to a boil, and preheat the grill.

Cut the stems off the artichokes, leaving a ½-inch length of stem. Using a serrated knife, cut off the tops of the artichokes. Peel off the thick leaves at the bottom of the artichokes. Stand the artichokes up with the cut tops on the surface and stems up, and cut them in half from stem to bottom. Using a small paring knife, cut out the fuzzy choke from the center of each artichoke half, and remove the tiny leaves in the centers. 

Place the artichokes into the boiling water and cook for about 12 minutes, until they are crisp-tender. Drain the artichokes and pat them dry. 

Pour the olive oil into a small bowl, and stir in the salt and pepper. Brush the artichokes with the seasoned olive oil on all sides. Place the artichokes on the grill cut side up, close the grill, and cook 5 minutes. Turn the artichokes, so that the cut sides are on the grill, and cook another 5 to 7 minutes, until the leaves are tender and they begin to char.

While the artichokes grill, make the aioli. In a small bowl, whisk together the mayonnaise, garlic clove, and lemon juice. 

Transfer the artichokes to a platter. Serve with lemon wedges and aioli alongside.