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.