Make the Lemon Sauce, if using: In a small bowl, stir together the sour cream, lemon juice, olive oil, tarragon, lemon zest, salt, and pepper. Refrigerate until ready to use (the sauce can be made up to a day ahead of time).
Using a large sharp knife, cut the top 1/2 inch from the artichokes. Trim the bottom 1/2 inch from the stems. Use kitchen shears to snip the pointy tips of the leaves off. With a large sharp knife, carefully slice the artichokes in half lengthwise, through the heart and the stem. Use a spoon to scrape out the thistle-y choke (the prickly hairy part that sits on top of the heart). As soon as you finish preparing each artichoke, rub all of the exposed cut sides with one of the lemon halves, which will prevent browning.
In a large pot, bring 1 inch of salted water to a boil. Place the artichokes in the pot and cover the pot. Allow the water to return to a simmer and steam the artichokes for about 15 to 20 minutes until fairly tender — a sharp knife should slide fairly easily through the stem, but they should not be too soft. Remove the artichokes from the pot and gently squeeze out any excess water with tongs. Or wait until they are cool enough to handle and gently squeeze out any excess water using your hands.
Heat the grill to medium. Brush the artichokes with olive oil and lightly salt them. Place the artichokes cut side down on the grill. Grill for 3 to 4 minutes, until you have nice grill marks on the bottom sides. Flip the artichoke with tongs, and grill for another 3 to 4 minutes until there are grill marks on the other sides, and until a sharp knife slides easily slides into the artichokes. Remove from the grill, and place cut sides up on a serving platter
Serve the artichokes with a sprinkle of fresh parsley and the lemon wedges. Serve the lemon sauce on the side.
Notes
Artichokes are very dense, and it’s very hard to get larger artichokes to become tender when cooking them only on the grill over direct heat. The best way to get tender yet smoky artichokes is to steam them first until they are basically tender, then finish them on the grill. You get those lovely grill marks, and there will be that appealing smoky flavor from the live fire. However, the bottom of the leaves and the heart will be completely tender if you cook them first.