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Adjust oven rack 6 inches from broiler element and heat broiler.
Line rimmed baking sheet with foil and spray with vegetable oil spray.
Brush eggplant slices with 2 tablespoons oil, sprinkle with ¾ teaspoon salt, and arrange on prepared sheet.
Broil eggplant until softened and beginning to brown, 10 to 14 minutes, flipping halfway through cooking.
Transfer eggplant to plate and set aside.
Pulse olives, parsley, red peppers, vinegar, garlic, pepper flakes, and remaining ¼ teaspoon salt in food processor until chopped, 8 to 10 pulses.
Transfer to bowl and stir in carrots and 2 tablespoons oil.
Arrange rolls on now-empty sheet and brush cut sides with remaining ¼ cup oil.
Broil rolls, cut sides up, until golden brown, 2 to 5 minutes.
Spread carrot-olive mixture evenly on cut sides of each roll.
Distribute mozzarella and eggplant evenly among roll bottoms, then cap with roll tops.
Serve.