Cut a little of the very top and bottom off of the artichokes. Snip the sharp points off of the leaves and rinse the artichokes. Cut each artichoke in quarters vertically and with a sharp paring knife cut the thistle out of each quarter.

Lay out a large piece of heavy duty aluminum foil and put the artichoke quarters in the center. Sprinkle a little water (about a tablespoon) over them. Add the lemon quarters. Cut the top off of 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 tightly around the packet so that some steam will build up while cooking.

Put on the barbeque, close the lid and cook for about 1 hour. Alternately you can cook them in the oven instead; put in a preheated, 400 degree oven for 1 hour.

Carefully open the foil packet and put artichokes on a serving platter. I toss the lemon wedges on the platter too.

Squeeze the garlic into some mayonnaise and mix well if you like for a delicious dipping sauce.