Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat.
For the court bouillon: Combine the garlic, dill, 4 cups of water and wine in a large stock pot. Squeeze the lemon juice into the pot, then add the lemon half. Season with salt and pepper. Bring to a gentle boil over direct heat.
For the artichokes: Cut off the top inch and most of the stem close to the base. Trim the tip of each outer leaf with kitchen shears. Trim the bottoms of the artichokes and prick with a fork or paring knife. Place over indirect heat to take on some smoke while the poaching liquid comes to a boil, about 5 minutes.
Place the artichokes in the liquid, bottoms up. Cover and simmer until a knife inserted into the bottom of the artichoke gives no resistance, about 30 minutes.
Meanwhile, make the vinaigrette: Whisk all the ingredients together in a medium bowl and season with salt and pepper.
Serve the artichokes with the vinaigrette on the side for dipping.
(Alternatively, you can poach the artichokes over medium-high heat on a stovetop.)