Prepare the Eggplant: Preheat the broiler to high with a rack 6 inches (15 cm) from the element. Line a large, rimmed baking sheet with heavy-duty foil (do not use parchment). Wash and dry the eggplants, halve lengthwise, and score the flesh in a deep diamond pattern without cutting through the skin.
Create the Marinade: In a small bowl, whisk the olive oil, za’atar, and sumac. Brush generously over the scored surface of each eggplant half, letting it seep into the cuts.
Char the Eggplant: Arrange halves cut-side up on the prepared sheet. Broil 5–7 minutes, watching closely, until well browned and lightly charred at the edges.
Bake Until Tender: Reduce the oven to 200°C (400°F). Move the sheet to the middle rack and bake 20–25 minutes, until the flesh is very tender and easily pierced with a fork.
Toast the Almonds: In a dry skillet over medium heat, toast the slivered almonds 2–3 minutes, stirring, until golden and fragrant; transfer to a plate to cool.
Blend the Sauce: In a food processor or blender, combine the Greek yogurt, tahini, lemon juice, garlic, and salt. With the machine running, slowly stream in the ¼ cup water until smooth and pourable; thin with more water, 1 tablespoon at a time, if needed.
Assemble and Serve: Let the eggplants cool a few minutes, transfer to a platter, drizzle generously with the sauce, and garnish with toasted almonds, mint, and parsley. Serve immediately.