Prepare the Eggplant: Preheat your oven's broiler to high with a rack positioned about 6 inches (15 cm) from the heating element. Line a large, rimmed baking sheet with parchment paper. Wash and dry the eggplants, then slice them in half lengthwise. Using a sharp knife, carefully score the flesh of each half in a deep diamond pattern, cutting down close to the skin but not through it.
Create the Marinade: In a small bowl, whisk together the extra virgin olive oil, za'atar, and sumac until thoroughly combined. Use a pastry brush to generously coat the entire scored surface of each eggplant half, allowing the marinade to seep into the cuts.
Char the Eggplant: Arrange the marinated eggplant halves, cut-side up, on the prepared baking sheet. Place the sheet under the preheated broiler for 5-7 minutes, watching closely until the tops are beautifully browned and slightly charred at the edges. This step develops a deep, smoky flavor.
Bake Until Tender: Remove the baking sheet from the oven. Reduce the oven's temperature to 200°C (400°F). Return the baking sheet to the middle rack of the oven and bake for an additional 20-25 minutes, or until the eggplant flesh is completely tender, creamy, and easily pierced with a fork.
Blend the Sauce: While the eggplant bakes, prepare the sauce. In a food processor or blender, combine the Greek yogurt, tahini, fresh lemon juice, minced garlic, and sea salt. Pulse a few times to combine. With the machine running, slowly stream in the ¼ cup of water until the sauce is smooth, creamy, and pourable. If it's too thick, add more water one tablespoon at a time to reach your desired consistency.
Assemble and Serve: Once the eggplants are tender, remove them from the oven and let them cool for a few minutes. Transfer the halves to a serving platter. Drizzle generously with the yogurt-tahini sauce. Garnish with the toasted slivered almonds and a scattering of fresh, chopped mint and parsley. Serve immediately.