Slice the eggplant lengthwise into slices one centimeter thick. Sprinkle both sides with salt and let them sit on paper towels for ten minutes. This draws out moisture, which helps the slices brown beautifully later instead of steaming in their own juices.
Pat the slices dry, brush them with two tablespoons of olive oil, and grill them in a smoking hot grill pan for three minutes per side, until they have dark grill marks and are completely soft. Undercooked eggplant remains spongy and bitter, so make sure they are fully cooked through.
Whisk the tahini with the juice of half a lemon, the grated garlic, a pinch of salt, and two to three tablespoons of cold water until smooth. The tahini will seize and split at first, then suddenly turn creamy; just keep stirring.
Rinse and drain the chickpeas, then mix them with the cumin, the remaining olive oil, some lemon juice, salt, and pepper.
Arrange the arugula on a plate, drape the grilled eggplant over it, and spoon the chickpeas in between. Generously drizzle the tahini sauce over everything, sprinkle with the pomegranate seeds and sumac, and finish with plenty of torn parsley and mint.