Prepare a stove-top griddle or outdoor grill. Snap off the tough outer leaves of the artichokes and discard. Rub the exposed surfaces with a lemon half. Cut off about the top third of each artichoke with a sharp knife. Trim the ends of the leaves with scissors. Trim all but 1/2 inch of the stem. Using a stainless-steel knife or a curved "bird's beak" knife, pare the remaining stem.
Steam the artichokes for 20 minutes. When cool, halve each and, using a melon baller, scoop out the prickly hairs, or the choke, and keep scraping until no more fuzz is apparent. Brush the halves with olive oil.
Grill each side for 5 to 7 minutes, until the artichokes are nicely browned. Spoon the tomato compote into each half and serve.