To grill the eggplants, preheat a cast iron griddle or gas or charcoal grill until hot. If using the oven, preheat to 450 degrees F. Pierce the eggplants all over with a fork. Place directly onto the griddle or grill, or onto a baking sheet for the oven. Cook, turning, until the entire surface is blackened and the flesh is soft, checking often. The timing will vary depending on which cooking method you use.
Immediately split open the hot eggplants and scoop the flesh into a bowl. This is important for keeping the color light. Discard the charred skin. If not using right away, the flesh can be kept refrigerated or frozen until needed. Do not drain any accumulated juices.
Let the eggplant cool, then mash by hand. Remove any firm pieces and set aside. Place the garlic in a food processor and process until finely chopped. Add the firm pieces of eggplant along with the tahini, lemon juice and salt. Process until completely smooth. Transfer the mixture to the bowl of eggplant and stir well. Taste and adjust the seasoning if needed.
Spoon onto plates or shallow bowls. Drizzle with oil. If desired, dust with cayenne. Serve with pita. It's also delicious as a dip for fried cauliflower or french fries.