Sprinkle cut sides of roll with balsamic vinegar. Layer one slice of mozzarella cheese, tomato slices, basil leaves, and the remaining slice of mozzarella cheese on the roll. Close sandwich; rub outside with olive oil.
Place sandwich in preheated skillet; top with another heavy skillet to press. Cook until bread is toasted and golden, about 3 minutes. Flip sandwich; top with skillet. Cook second side until toasted, about an additional 2 minutes.