Place lemons in a large bowl of cold water. Cut the very top and bottom off the artichokes. Snip the sharp points off the leaves with scissors and rinse the artichokes. Cut into quarters vertically. With a sharp paring knife or spoon, cut the thistle out of each quarter. Drop the artichokes into the lemon water to prevent browning.

Prepare the grill for indirect heat and preheat to medium.

Lay out a large piece of heavy-duty aluminum foil. Remove the artichokes from the water, shaking off the excess water. Place in the center of the foil along with the lemon wedges. Cut the top off the head of garlic and toss that in. Sprinkle with salt and pepper and wrap the foil tightly around the whole lot. Take another large piece of foil and wrap it tightly around the packet so steam will build up while cooking. Place on the grill, close the lid, and cook over indirect heat for 50-60 minutes or until the leaves of the artichokes pull off easily.

Carefully open the foil and put the artichokes and lemon wedges on a serving platter.

Squeeze the garlic cloves into the mayonnaise and mix. Serve as a dipping sauce for the artichokes.