1. Set the grill up for indirect and direct cooking over medium heat (375° to 425°).
2.  Cut each eggplant in half lengthwise and lightly score the flesh of each eggplant making sure not to cut through the skin. Drizzle each half with olive oil and season with salt, pepper, thyme and za’atar and let it absorb into the eggplant for 5-10 minutes.
3.  Mix the sauce ingredients together and set aside in the fridge for service.
4.  Grill each eggplant cut-side down over the direct heat for 3-5 minutes then move them all to the indirect side and flip them cut-side up for another 5-7 minutes until the insides are soft and creamy.
5.   Remove from the grill and serve with the buttermilk sauce spooned overtop of the cut side of the eggplant, with a pinch of za’atar and lots of pomegranate seeds and parsley scattered.