Fill a chimney starter to the rim with charcoal and burn the charcoal until it is lightly covered with ash. Spread the charcoal in a tightly packed, single layer across one-half of the charcoal grate. Let the coals burn down to medium heat (180°C to 230°C). Leave all the vents open.
Lay the onion halves, cut-sides down, on a 40 centimetre square of heavy-duty aluminum foil. Nestle the garlic cloves between the onions. Drizzle 2 tablespoons of oil over the onions and garlic. Fold up the sides of the foil and seal the packet. Place the packet right on top of the charcoal and cook, with the lid closed, until a knife inserted through the foil slides easily in and out of the onions, 20 to 30 minutes, carefully turning the packet with tongs once or twice. Carefully remove the packet with tongs and let cool to room temperature.
Lightly brush the pita bread pockets on both sides with oil. Set the cooking grate in place and brush clean. Working in batches of three or four pitas at a time, grill them over direct medium heat, with the lid open, until crisp and toasted, 2 to 4 minutes, turning frequently and swapping their positions as needed for even cooking.
