Slice tomatoes about 1/4 inch thick (you should get at least 4 slices per tomato). Chop excess tomato and reserve to garnish serving plate.
To assemble the "stacks", place 1 slice of tomato in center of serving plate. Top tomato slice with 1 fresh basil leaf. Add slice of mozzarella, then another fresh basil leaf. Finish by topping with second tomato slice. Place decorative basil leaf on top. Garnish plate with remaining chopped tomato and torn basil leaves. Before serving, drizzle olive oil and balsamic vinegar over the top.