Soak the wood chips in water for at least 30 minutes.
Prepare the grill for indirect cooking over medium heat (350° to 450°F).
Lightly brush the tomatoes with oil. Cut a sheet of aluminum foil about 8 by 12 inches. Put the garlic cloves in the center of the foil. Drizzle ½ teaspoon of oil over the garlic cloves and fold up all four sides to make a packet.
Drain and add the wood chips directly onto burning coals or to the smoker box of a gas grill, following manufacturer’s instructions. Brush the cooking grates clean. As soon as the wood begins to smoke, place the tomatoes and the packet of garlic over indirect medium heat. Cook until the tomatoes are lightly smoked and browned in spots, and the garlic is browned along the edges, about 20 minutes.
In the bowl of a food processor, puree the tomatoes and garlic until they are smooth. Add the mayonnaise, vinegar, ½ teaspoon of the salt, and the pepper. Pour the sauce into individual serving bowls. (Cover and refrigerate if you are not planning to serve the sauce within the next hour. Let stand at room temperature for about 30 minutes before serving.)
Bring a large pot of salted water to a boil on the stove.
Prepare the artichokes by snapping off the dark outer leaves until you expose the yellowish leaves with pale green tips. Lay each artichoke on its side and cut off the remaining leaves just above the base. Using a teaspoon, scoop out the fuzzy choke. Cut the base of each artichoke in half lengthwise, through the stem, and then trim the stem, leaving about ¾ inch attached. Using a small, sharp knife or vegetable peeler, pare away the dark green skin around the base and from the stem to expose the tender lighter green flesh. After each artichoke heart is trimmed, place it into a large bowl of water mixed with lemon juice.
Related Grill Skills
Prepping Artichoke Hearts
Drain the artichokes. Add them to the pot of boiling water and boil until you can pierce them easily with a thin knife, 10 to 12 minutes, being careful not to overcook them. Drain the artichokes in a colander and place them in a large bowl. While still warm, add 2 tablespoons of oil and the remaining ½ teaspoon of salt. Toss gently to coat the artichokes. (The artichokes may be made up to this point and refrigerated for up to 4 hours. Bring to room temperature before grilling.)
Lift the artichoke hearts from the bowl letting any excess oil drip back into the bowl. Grill the artichokes over direct heat, with the lid closed, until warm and lightly charred, 4 to 6 minutes, turning them once or twice. Serve warm or at room temperature with the sauce.