Preheat the oven to 350 degrees F and preheat a charcoal grill and let the coals burn down to a gray ash. (Alternatively, preheat a broiler and lightly oil a broiling pan.)
Brush the eggplant slices with the 1/2 cup oil and season both sides with salt and pepper. Grill or broil on a rack set 5 to 6 inches from the heat, turning once, until tender, about 5 minutes. Transfer to a baking sheet to cool. Brush the onions with the remaining 2 tablespoons oil and season with salt and pepper. Grill or broil until brown and tender, about 5 minutes. Wrap the tortillas in foil and heat in the oven for 5 minutes.
Whisk the mayonnaise with the garlic and lemon juice in a small bowl to make the aioli. Spread 2 tablespoons of the aioli over each tortilla. Cover the bottom 1/3 of each tortilla with 1/4 of the eggplant and 1/4 of the onions. Sprinkle on a 12/4 of the cheese and top with mint leaves. To make the wrap, fold in the two sides of the tortilla. Roll away from you, tucking in the edges to form a tight cylinder. Wrap the cylinder in foil and cut in 1/2 diagonally right through the foil. The foil becomes the holder to be peeled away as the wrap is consumed. Serve the wrap hot or at room temperature.
Sara Moulton Cooks at Home, Broadway Books, 2002