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 (350° to 450°F). Leave all the vents open.
Lay the onion halves, cut sides down, on a 16-inch square of heavy-duty aluminum foil. Tuck 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, turning the packet with tongs once or twice. Remove the packet with tongs and let cool to room temperature.
Lightly brush the pita bread pockets on both sides with oil. Brush the cooking grates 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.
Open the packet and remove and discard the skins and hard stem ends from the onions and garlic. Place the roasted onions and garlic in the bowl of a food processor fitted with a metal blade. Pulse until almost pureed. Add all the remaining ingredients and process until quite smooth. Cut each pita into wedges. Serve with the dip.