Preheat the oven to 450˚. Toss the eggplants in a large bowl with 3 tablespoons olive oil, 1/2 teaspoon salt and a few grinds of pepper. Lay the eggplant slices in a single layer on a baking sheet. Roast until dark brown, flipping halfway through, 25 to 30 minutes.
Meanwhile, bring a small saucepan of water to a boil. Gently add the eggs and cook, adjusting the heat to maintain a gentle boil, 9 minutes. Remove with a slotted spoon to a bowl of ice water. Let stand until cool, about 5 minutes. Peel and slice the eggs.
Whisk the tahini, 1/3 cup warm water, 2 tablespoons lemon juice and 1/4 teaspoon salt in a medium bowl until smooth. Add 1 more tablespoon of water, if needed, to loosen.
Halve the tomatoes lengthwise, remove the seeds, chop and put in a medium bowl. Chop the cucumbers and parsley and add to the tomatoes along with the remaining 1 tablespoon each olive oil and lemon juice, 1/4 teaspoon salt and a few grinds of pepper; toss well.
Warm the pitas directly on the oven rack, about 1 minute. Spread the pitas with the hummus, then drizzle with some of the tahini sauce. Top with the eggplant and eggs; season with salt and pepper. Top with the tomato-cucumber salad, the remaining tahini sauce and the banana peppers.