Heat the broiler to high (500 F / 260 C) and set a rack about 4-5 inches from the element. Line a sheet pan 33 x 46 cm / 13 x 18 inch with foil.
Pierce each eggplant a few times with a fork and place them on the sheet pan. Broil for 20 to 25 minutes, turning every 5 to 7 minutes, until the skins are blistered and blackened on all sides and the flesh feels completely collapsed when pressed with tongs.
Transfer the eggplants to a colander set over a bowl. Let them cool and drain for 5 to 10 minutes; this gets rid of the bitter, watery juices that pool inside a charred eggplant.
Split the eggplants open and scoop the flesh into a mixing bowl, leaving the blackened skins behind. Chop the flesh roughly with a knife instead of pureeing it smooth, so some texture stays.
In a separate small bowl, whisk together the tahini, lemon juice, garlic, olive oil, cumin, and salt until you get a smooth, pale sauce.
Fold the chopped eggplant into the tahini mixture until combined but still a bit chunky. Taste and add more lemon juice or salt as needed.
Spread the dip onto a serving plate, making a shallow well in the center. Drizzle with olive oil and scatter over the pomegranate seeds, parsley or mint, sumac, and smoked paprika.
Serve at room temperature or slightly chilled with warm pita or raw vegetables.