Slice artichoke tops off crosswise to remove tips; discard. Pull small leaves from artichokes. With scissors, cut off and discard thorny tips from remaining outer leaves. With a knife, trim fibrous exterior from bottoms and stems.
Half-fill an 8- to 10-quart pan with water. Cover and bring to a boil over high heat.
Add artichokes, cover, and simmer until bottoms pierce easily, about 20 minutes.
Drain artichokes and let stand until cool enough to handle.
Cut each one in half lengthwise and scrape out and discard fuzzy center.
In a large bowl, combine vinegar, 1/4 cup water, dill, soy, ginger, garlic, and oil. Spoon half of the vinegar sauce into a small bowl.
Roll artichokes in large bowl of sauce.
Lift artichokes from bowl, draining, then lay them, cut side down, on a barbecue grill over a solid bed of medium coals or a gas grill on medium (you can hold your hand at grill level only 4 to 5 seconds); close lid on gas grill.
As artichokes cook, baste every few minutes with sauce from large bowl. Grill until lightly browned on bottom, 5 to 7 minutes. Turn artichokes over and spoon remaining basting sauce into them. Grill until leaf tips are lightly charred, 3 to 4 minutes more.
Arrange artichokes on a platter. Break off leaves and cut up bottoms to dunk into the small bowl of sauce.
