Heat oven to 400ºF. On a parchment-lined baking sheet, toss eggplant with a drizzle of oil and generous pinch salt. Arrange in a single layer and roast until golden and tender, 20-30 minutes. Let cool then cut into 3 to 4 pieces each.
In a bowl, stir together tahini and pinch salt, adding cold water little by little to reach the consistency of maple syrup. Adjust salt to taste.
In a second bowl, stir together parsley, scallion, preserved lemon paste, and a drizzle of olive oil. Adjust preserved lemon paste to taste.
Spread matbucha on a serving plate. Top with eggplant pieces, drizzle with tahini sauce, then arrange eggs on top and sprinkle with parsley mixture. Drizzle with olive oil and serve.
