Separate the cloves of garlic, but do not peel them. Place in a small glass pyrex dish and drizzle with the olive oil. Cover with foil and place in the oven to roast until caramelized and tender (anywhere from 30 minutes to 45 minutes).
Slice the top off of each of the eggplants and discard. Using an open gas flame, set the flame/heat level to medium and place one of the eggplants directly over that with heat-resistant tongs. Leave it there for 2-3 minutes—then turn to another side to char for 2-3 minutes more. Repeat this until all sides are charred.
By this time, the eggplant will start to sputter slightly, but keep turning until the entire skin is completely charred and wrinkled. (It should take 13-15 minutes to achieve a good char all over.)
Place the eggplant on some absorbent towels to slightly drain while you prepare the second eggplant. When both eggplants are done, you can half them and scoop/scrape out the insides with a spoon, then discard the skins. Place the eggplant into a small food processor.
By this time, your garlic should be almost ready. Remove from the oven and remove the cloves to cool. Pour the rich and garlicky leftover oil into the food processor with the eggplant. Once the garlic is cooled, add it to the food processor, along with the next four ingredients. Blend until smooth.
You can either serve immediately or refrigerate for a day or two before serving.