Prepare the Lemon Vinaigrette; set aside until ready to use.
Bend back lower, outer petals of artichokes until they snap off easily near base. Continue doing this until you reach a point where the leaves are upright and tightly closed.
Using a sharp stainless steel knife, cut off top third of artichokes or just below the green tips of the petals. Pare all remaining dark green areas from bases. Cut off stems.
Halve or quarter as desired. If center petals are purple or pink remove center petals and fuzzy centers. Dip or rub all surfaces with lemon juice.In a large saucepan, bring 1 1/2 quarts of water to a boil. Add prepared baby artichokes and cook approximately 7 to 10 minutes or until you can easily pierce them with a fork, but they still offer some resistance. Drain and immediately and immerse in cold water to stop the cooking.
When cool, cut the baby artichoke in half lengthwise, sprinkle them with olive oil, and salt and pepper to taste.
Prepare barbecue grill. Cover barbecue with lid, open any vents, and grill them over a medium-hot fire, cut side down for about 5 minutes, or until the cut sides are well browned.
Pour the Lemon Vinaigrette over grilled artichokes and toss. This can be served right away, but it is much better if allowed to sit for an hour for the flavors to mingle. It will keep, covered and refrigerated, for about 3 days.