Slice the rolls in half and use 1 tablespoon of the olive oil to brush the insides of the rolls. On the bottom half of each, layer 1/4 of the shredded chicken, a slice or two of the mozzarella, a slice of tomato, and 2 to 3 basil leaves; sprinkle with salt. Put the top of the roll on the sandwich and press down gently.
Put the remaining olive oil in a skillet large enough to hold all 4 sandwiches; turn the heat to medium. When the oil is hot and shimmers, put the sandwiches in the pan and put another pan or large heatproof plate on top of the sandwiches and press down (you can also use an electric sandwich press or waffle iron for this step if you like). When the cheese has melted and the bread begins to crisp on the bottom, about 3 to 4 minutes, remove the top pan or plate, flip the sandwiches, press down again, and cook for another 2 or 3 minutes. Serve immediately.