Heat oven to 375°F. Halve eggplants lengthwise, place cut side up on a parchment-lined baking sheet, drizzle with olive oil, and season with salt. Bake for 30 to 40 minutes, until eggplant is lightly browned. Remove from oven, and cool to room temperature.
While eggplant is cooking, heat 1 tablespoon oil in a pan over low heat. Add onions and cook, stirring occasionally, until soft and translucent, about 10 minutes. Add garlic cloves and cook for another 5 minutes. Remove from heat and set aside.
Once eggplant is cool enough to handle, scrape flesh into a food processor fitted with the steel blade. Add the onion-garlic mixture, tahini, lemon juice, and labneh, and pulse 5 to 10 times. Add paprika, Aleppo pepper, Urfa chile, and pomegranate molasses, and pulse until thick and evenly combined. Taste and season with salt and pepper as needed.