First make the dressing: pack the basil leaves into the bowl of a food processor and add the lemon juice, salt and pepper. With the machine running, gradually add the oil through the lid to make a smooth dressing. Adjust the seasoning to taste.
Cut the avocados in half and remove the stones. Carefully peel the skin, then cut each half across into half moon shaped slices.
Slice the mozzarella cheese and the tomatoes thinly. Arrange the avocado, mozzarella and tomato slices overlapping on 4 individual plates, then drizzle over the dressing. Sprinkle with a little more salt and pepper and serve with the ciabatta bread.