TO PREPARE THE DIP
In a bowl, combine together the yogurt, parsley, tarragon, chives, salt, pepper and lemon juice and stir to mix well. Taste and adjust seasonings as desired.
TO PREPARE THE ARTICHOKES
Working with one artichoke at a time, trim the stem even with the base. Snap off the small, tough leaves around base and trim the stub ends flush. Using a very sharp knife, cut off the upper third of the artichoke. Rub the cut surfaces with the lemon half to prevent browning.
Pour water into a steamer pan to a depth of about 3 inches, put the rack in place and bring the water to a boil over high heat. Place the artichokes, stem end up, on the rack, then reduce the heat to medium, cover and steam until the base of an artichoke offers little resistance when pierced with the tines of a fork, about 20 to 30 minutes. The timing will depend on the size and maturity of the artichokes.
Remove the artichokes from the steamer and set aside until cool enough to handle. With a sharp knife–I like to use a serrated one–cut the artichokes in half lengthwise. Then, using a spoon, scoop out the central leaves from each artichoke, removing the thistles and any furry bits.
In a bowl large enough to hold all the artichoke halves, combine the parsley, garlic, salt, pepper, lemon juice and olive oil. Add the artichokes, then gently turn several times to coat well. Set aside to marinate for at least 20 minutes, and up to 3 to 4 hours.
Prepare a fire in a charcoal grill or preheat a gas grill. When hot, place the artichokes, cut side up, on the grill and allow to sear, turning them to reach all the outer leaves, about 3 minutes total. Turn and sear the inside, about another 3 minutes. Remove to a platter and let cool to room temperature. Serve accompanied by the green herb dip.