Lay mozzarella on paper towels for 10 minutes and blot dry. Salt tomato slices, let them stand for 10 minutes alongside the cheese, then blot away released liquid.
Divide half the mozzarella over the bottom ciabatta halves. Add one flat tomato layer, sliced basil and black pepper, then the remaining mozzarella and top halves.
Brush the outside faces lightly with olive oil. Heat a large skillet over medium-low heat.
Cook sandwiches for 4 to 5 minutes per side, pressing gently with a spatula and covering for 2 minutes after turning, until the crust is golden and mozzarella is soft throughout.
Rest for 1 minute and cut each sandwich across the middle before serving.