Trim ends off eggplant; discard. Cut eggplant lengthwise into 1/4-inch-thick slices, discarding the outside "peel" slices. Spread the eggplant slices in a single layer on a baking sheet. Sprinkle with 1/2 teaspoon of the salt; cover and set aside for 30 minutes. Rinse eggplant; pat dry. Rub eggplant slices with half of the olive oil; set aside.
For pepper spread, in a large skillet heat remaining oil. Cook onion and garlic, uncovered, in hot oil over medium heat for 5 minutes or until onion turns golden, stirring often. Add the cherry or banana peppers, the balsamic vinegar, and the remaining 1/4 teaspoon salt; cook and stir for 2 minutes. Reduce heat; cover. Cook over low heat for 10 minutes or until mixture is soft, stirring occasionally. Remove pepper mixture from heat. Let cool.
In a food processor bowl or blender container, cover and process or blend pepper mixture until nearly smooth. Set aside.
Score meat on both sides by making shallow diagonal cuts at 1-inch intervals in a diamond pattern. Season steak with salt and pepper, if desired. Rub steak on both sides with dried Italian seasoning.
Grill steak on rack of uncovered grill directly over medium coals for 18 to 22 minutes for medium doneness (160 degrees F), turning once. Grill eggplant slices for 6 to 8 minutes or until cooked through, turning once. Set aside. If desired, brush cut sides of Italian bread halves with olive oil. Grill bread, cut side down, for 1 to 2 minutes or until toasted.
To assemble, spread about 1 tablespoon pepper spread on bottom half of each Italian roll. Top with greens and eggplant slices. Slice steak across grain into very thin strips. Arrange sliced steak on eggplant slices. Add Parmesan, if desired. Lightly spread top halves of Italian rolls with remaining pepper spread. Serve open-face with top halves of rolls on the side. Garnish with sprigs of thyme, if desired. Makes 4 servings.