Preheat the grill , if using, or the broiler.
If using the broiler, line a baking sheet with aluminum foil (for easy clean-up) and place a rack in the prepared pan.
Using a fork, pierce the eggplants all over, several times, and arrange them on the rack of the pan or on the grill.
Wrap the garlic cloves in some foil and place them on the rack of the pan or on the grill.
Cook the garlic for about 15 minutes. Remove and set aside.
If using the grill, roast the eggplants for about 30 minutes on the grill, turning them occasionally until the eggplants are soft (when pierced with a fork) and their skin is charred on all sides evenly.
If using the broiler, roast the eggplants for about 40 minutes, turning them occasionally until the eggplants are soft (when pierced with a fork) and their skin is black on all sides evenly.
When ready, set them aside for about 10 minutes to cool.
Place a mesh strainer over a large mixing bowl.
Cut the eggplants lengthwise scoop out the flesh and transfer to the strainer. If the eggplants have a lot of large seed, remove and discard the part with the seeds. Discard the skins.
Let rest for 15 minutes. Using a spoon, press occasionally to drain as much liquid as possible. Discard the juice as it is very bitter.
Wipe dry the bowl before adding the eggplant mixture to the same bowl and set aside.
Squeeze the roasted garlic cloves to release the edible interior into the bowl of a food processor.
Add the eggplant, tahini, olive oil, lemon juice, salt and pepper, and process until creamy.
Add the tahini, lemon juice, parsley, and one-quarter (4 tbsp) cup of olive oil into the bowl.
Transfer the dip back to the mixing bowl, add the parsley and stir to combine.
Cover and refrigerate for at least one hour, to allow the flavors to blend.
Transfer the dip to serving a bowl and garnish with the remaining olive oil and the pomegranate arils.
Serve with a selection of vegetable crudités and/or warmed pita wedges.