Arrange an oven rack in the middle position and preheat the oven to 475 degrees F. Preheat a baking sheet in the oven for 10 minutes. Toss the eggplant rounds evenly with 3/4 teaspoon salt and a few grinds of pepper in a large bowl; add 1/4 cup olive oil and toss again, making sure the eggplant soaks up all the oil. Remove the hot baking sheet and drizzle with 1 tablespoon olive oil; lay out the eggplant slices in one layer. Roast until soft and golden to dark brown on the underside, about 20 minutes. Let cool. Blot dry by pressing lightly with a paper towel.
Meanwhile, put the chickpeas in a medium bowl. Mash with a fork or a potato masher until coarsely mashed with lots of large pieces. Add the lemon zest and juice, red onion, mint, 1/2 teaspoon salt, a few grinds of pepper and the remaining 2 tablespoons olive oil and mix to combine.
Assemble the sandwich by slicing the ciabatta loaf in half horizontally and lay the two halves cut-side up. Pull out some of the bread from both halves (save for another use), leaving a 1/2-inch border. Spread the hummus on both the bottom and top halves of the bread. On the bottom half, continue layering with half the chickpea mixture, the eggplant, and then the remaining chickpea mixture. Top with the cucumber, pepperoncini, shredded carrot, and the top half of the bread. Wrap tightly in parchment, wax paper or plastic wrap and tie with twine if necessary. Press 3 hours with a brick (see Cook's Note) or cast-iron skillet on top to flatten the sandwich, refrigerated. About 45 minutes before serving, bring the sandwich to room temperature.
A brick that weighs about 4 pounds works best. Wrap it in foil to protect kitchen surfaces, if desired.
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