Preheat the grill and GrillGrates to a moderate low temperature, about 300 degrees F (150 degrees C).
Cut the Ciabatta rolls in half and open up.
Layer with the sliced tomato, mozzarella cheese, basil, and a sprinkle of balsamic vinegar. If you have reduced balsamic vinegar this can work better as it is thicker and less likely to make the roll go soggy.
Close the sandwich then place it on the preheated GrillGrates. Place an upside-down skillet over the top to keep the heat focused on the panini.
After about 5 minutes, or when the roll has brown grill marks on it, remove the skillet, turn the panini, then replace the skillet to finish the cooking, about 5 more minutes.
Halve and serve immediately while the cheese is still gooey.