Put the tahini, ice cubes, garlic, lemon juice, and ½ teaspoon salt in a food processor and process until smooth. With the machine running add the water, a little at a time until the paste is thinned out and almost white. The consistency should be similar to a dip. Can be made up to a month ahead and stored in an airtight container in the refrigerator.
Brush the olive oil on both sides of the eggplant slices. Sprinkle on both sides with za’atar and a pinch of salt and pepper. Heat a grill pan or skillet over medium heat. Cook the eggplant in a single layer for 4-5 minutes per side or until browned and soft in the center. Can be made a day ahead and stored covered in the refrigerator.
To serve, spread the tahini on a platter, top with eggplant, drizzle with balsamic glaze or pomegranate molasses, and top with pomegranate seeds and chopped parsley. Can be served cold, warm, or at room temperature