Preheat oven to 400 degrees F (200 degrees C). For each artichoke, rinse under cold water. Pat dry with a clean kitchen towel or paper towel. Remove any stray leaves from the stem of the artichoke. With kitchen shears, or sharp scissors, remove the thorny tips from the leaves. With a sharp chef's knife or serrated knife, cut about an inch off of the top of the artichoke. Keep a lemon handy to rub onto the exposed areas so that they do not oxidize and turn brown. Remove the bitter, fibrous end of the stem with your knife, leaving about an inch left on the artichoke. Be sure to rub a lemon onto the exposed end of the stem. Peel the outer skin from the remaining stem. The stem can has a more bitter taste than the rest of the artichoke and removing the skin helps to take away some of the bitterness. Rub exposed peeled stem with lemon.
Run the artichoke under cold water, pulling apart the leaves to carefully rinse out the vegetable and remove any impurities.
With a sharp chef's knife, cut the artichoke in half. Remove the furry, inedible choke with a melon baller or spoon. Rinse the artichoke to remove all pieces of the furry choke, if needed. Be sure to keep exposed areas covered with lemon juice throughout process. You can grill the artichokes as halves, or slice them into quarters. Quarters will cook a bit faster; halves make a nice presentation.