Line a baking sheet with aluminum foil; lightly oil the foil Cut the eggplant lengthwise into 1/2 inch thick slices. Place on the prepared baking sheet; brush with 1 tbsp olive oil. Broil 5 1/2 inches from the heat with the oven door partially opened for 6 to 7 minutes or until tender and golden -- watch it so it doesn't burn. Coarsely chop the eggplant and set aside.
Saute onion, garlic, black pepper, and thyme in the remaining 2 tbsp olive oil in a large skillet over medium-high heat, stirring constantly, until tender. Stir in the reserved eggplant, tomatoes, parsley, and vinegar. Remove from heat and set aside.
Cut the bread in half lengthwise; rub each cut side with the halved garlic clove; place on an ungreased baking sheet. Broil until lightly browned. Place the reserved eggplant mixture on the bottom half of the toasted bread; top with mozzarella cheese slices. Broil 5 1/2 inches from the heat 1 to 2 minutes or until the cheese melts. Cover with the top half of the bread.