With a sharp knife, cut off the stems, leaving about 1 inch, and cut the top off the choke straight across. With kitchen shears, trim the thorny tips (if any) off the leaves. Pull off any loose leaves from around the bottoms.
In a large pot over high heat, in 1 inch of boiling water, place the artichokes face down with their stems facing upward. Add a few lemon slices if you like and heat to boiling. Reduce heat to low; cover and simmer for 20-25 minutes or until a leaf can be pulled off easily or you can stick your fork easily into the middle of the choke.
When cool enough to handle, cut each in half lengthwise and scrape out the fuzzy center (the choke). In a bowl combine 1 clove garlic (or more), 1 tablespoon lemon juice, ½ teaspoon salt and pepper and olive oil. Brush artichokes with garlic mixture and set, cut side down on a grill over medium heat. Grill, turning once, until lightly browned, 5-6 minutes or until warm.
