Preheat a char-grill pan over high heat. Combine eggplant and olive oil in a bowl, season to taste and toss to combine. Cook half the eggplant, turning occasionally, until golden and tender (1-2 minutes each side). Transfer to a bowl, add garlic and repeat with remaining eggplant. Drizzle eggplant with vincotto or balsamic vinegar, then add tomato and herbs, toss to combine, season to taste and set aside.
Meanwhile, for rosemary bruschetta, dip rosemary sprigs in olive oil and brush bread slices on one side with oil. Char-grill bread, oiled-side down, in batches and, while cooking, brush other side with oil using rosemary sprigs and cook until golden (1-2 minutes each side). Rub with cut-side of halved garlic and keep warm.
Divide eggplant among plates, top with mozzarella or fior di latte, drizzle with extra-virgin olive oil and serve with bruschetta.