Preheat a gas or charcoal grill to medium (about 500 degrees). Place the eggplants on the grate directly over the heat source. Cook, turning occasionally with tongs, until completely tender and well charred on all sides, about 20–30 minutes. Remove to a cutting board and loosely tent with foil for 10 minutes.
Working with one eggplant at a time, use a sharp chef’s knife to slit it open lengthwise. Carefully scoop out the soft flesh with a large spoon and transfer to a fine-mesh strainer set in a large bowl. Once all the eggplant is scooped, pick out any stray bits of skin or blackened flesh and discard. Allow the eggplant to drain, gently pressing with a wooden spoon a few times, about 20 minutes.
Transfer the eggplant to a large bowl. Add the lemon juice and garlic, then stir vigorously with a fork until the eggplant breaks down into a rough paste. Stirring constantly and vigorously, add tahini, followed by the olive oil in a thin, steady stream. The mixture should become pale and creamy. Stir in the chopped parsley.
Transfer to a serving bowl. Drizzle the pomegranate molasses over the top and garnish with pomegranate seeds, green onion, and parsley leaves, if desired. Serve with grilled flatbread, crackers, and assorted vegetables such as carrot sticks, green beans, and radishes.
