Soak the wood chips in water for at least 30 minutes.
Prepare the grill for indirect and direct 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.
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 small 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, trim and smooth the rough and greenish areas around the base. Trim about 1⁄16 inch all the way around the stem to expose the tender part of the stem. After each artichoke heart is trimmed, place it into a large bowl mixed with lemon juice.
Cook the artichokes in the boiling salted water until you can pierce them easily with a 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 medium heat, with the lid closed as much as possible, until warm and lightly charred, 4 to 6 minutes, turning them once or twice. Serve warm or at room temperature with the sauce.
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