Wash large artichokes; trim stems, if desired, and remove loose outer leaves. Snip off the sharp leaf tips. If using baby artichokes, remove outer leaves to reach pale green or yellow leaves on bottom half. Cut darker green portion of leaves off top half of artichoke; discard. Cut off stem and trim any remaining green from base of baby artichokes.
In a Dutch oven bring a large amount of lightly salted water to boiling; add artichokes. Return to boiling; reduce heat. Simmer, covered, for 20 to 30 minutes for large artichokes or 10 minutes for baby artichokes or until a leaf pulls out easily.
Place artichokes in a large bowl of ice water to cool completely. Drain artichokes upside down on paper towels. Cut artichokes in half from top through stem; use a spoon to scoop out the fibrous cores, leaving the hearts and leaves intact.
For Tarragon Drizzle, in a screw-top jar combine the 2/3 cup oil, vinegar, green onions, mustard, and tarragon. Cover and shake well. Season to taste with salt and pepper.
Brush artichoke halves with additional olive oil. For a charcoal grill, place artichokes, cut sides down, on the rack of an uncovered grill directly over medium coals. Grill for 7 minutes; turn artichokes and grill for 5 to 7 minutes. (For a gas grill, preheat grill. Reduce heat to medium. Place artichokes on grill rack over heat. Cover and grill as above.)
On a serving platter place grilled artichokes cut sides up. Shake Tarragon Drizzle; pour some over artichokes; pass remaining with platter. Serve with lemon wedges. Makes 8 to 10 servings.