Prepare the grill and let burn down to medium coals or heat gas grill on medium heat.
In a colander set over another bowl, salt the eggplant and let drain for 30 minutes then rinse and dry.
Brush the eggplant slices with olive oil and season with pepper. Toss the peppers with a little olive oil. Place both on the grill and cook the eggplant until browned on the first side, about 3 minutes. The slices should be browned between the grill marks when ready to turn. Move them away from direct heat and cook on the second side until tender, about 10 minutes total. Cut the grilled slices into 1/2-inch dice.
Cook the peppers, turning as necessary, until charred all over. Remove the peppers to a bowl. Cover and let the peppers steam to loosen the skins. When cool enough to handle, peel, seed, and tear into long, thin strips.
Divide the dough into 6 equal balls. Working on a surface free of flour, roll each ball under your palm. As it rolls, it will stick slightly to the surface, creating tension that helps form a tight, round ball. Dust the work surface lightly with flour, pat each ball down lightly, dust the tops with flour, cover with a towel, and let rise for about 15 minutes. With a rolling pin, roll out each ball into a circle 8 or 9 inches in diameter and about 1/8-inch thick.
Have the grill ready at medium heat. Have the 1/4 cup olive oil, eggplant, peppers, garlic, oregano, red pepper flakes, if using, olives, and cheese in bowls and ready by the side of the grill.
Brush each dough round with about 1 teaspoon of the olive oil. Place on the grill, oiled side up, and cook until browned on the underside, about 2 minutes. Turn and, working quickly, brush each round with another 1 teaspoon olive oil. Then scatter each with 1/6 each of the garlic, oregano, red pepper flakes, if using, eggplant, bell peppers, olives, and cheese. Cover the grill and cook until the rounds are browned on the bottom and the cheese has softened, about 7 minutes. Transfer to plates and top each with an equal amount of the arugula. Spritz lightly with vinegar. Serve immediately.