Using a mortar and pestle or a food processor, puree the garlic into a coarse puree. Add the basil and pound until it is coarsely chopped. Add the pine nuts and smash to break them up.
Mix in the Parmesan cheese.
Slowly mix the olive oil into the pesto and grind until well blended. Squeeze the lemon juice into the pesto and stir to blend. Season the pesto to taste with salt and pepper and set aside.
Preheat the barbecue or a grill pan over medium heat. Using a serrated knife, cut the ciabatta loaf horizontally into 3 slices.
Lay the bottom and middle bread slices on the work surface. Arrange the avocado slices over the bottom and middle bread slices. Arrange the tomato slices over the avocado. Season with salt and pepper.
Tear the mozzarella cheese and scatter it over the tomato slices.
Spoon the pesto over the cheese.
Assemble the sandwich layers, positioning the middle layer atop the bottom layer. Cover with the top bread slice and press the sandwich firmly to compress. Brush the oil over the sandwich.
Wrap 2 bricks with aluminum foil.
Place the sandwich on the grill. Set a baking sheet atop the sandwich then weigh it down with the bricks. Grill the panini until the bread is toasted and the cheese has melted, about 5 minutes on each side.
Cut the panini into 4 pieces. Arrange the sandwiches on a platter and serve immediately.
Panino (singular); Panini (plural): Italian for “small bread,” referring to a sandwich or roll.
Mozzarella is a generic term for several kinds of originally Italian cheeses that are made using spinning and then cutting (hence the name; the Italian verb mozzare means "to cut"):
Mozzarella di Bufala (buffalo mozzarella), made from domesticated water buffalomilk. Mozzarella di bufala campana (PDO 1996) is a particular type of mozzarella, made from the milk of water buffaloes raised in designated areas of Lazio and Campania; some consider it the best for flavour or quality. Unlike other mozzarellas, 50% of whose production derives from imported, and often semi-coagulated milk, it is protected by European PDO. It is a raw material in Italian style Neapolitan pizza - rather than mozzarella made with pasteurized cow's milk. Mozzarella is available fresh; it is usually rolled in the shape of a ball of 80 to 100 grams (6 cm diameter), sometimes up to 1 kilogram (about 12 cm diameter), and soaked in salt water or whey, sometimes with added citric acid, until sold. It is a semi-soft cheese. Due to its high moisture content, it is traditionally served the day it is made, but can be kept in brine for up to a week, or longer when sold in vacuum-sealed packages.
Mozzarella fior di latte, made from fresh pasteurized or unpasteurized cow's milk Fior di latte (written also as fiordilatte) is used to distinguish the mozzarella made from cow's milk from that made from buffalo's milk.
Low-moisture mozzarella, which is made from whole or part skim milk, and widely used in the foodservice industry. When slightly desiccated (partially dried), the structure becomes more compact; then it is better used to prepare dishes cooked in the oven, for example lasagne. Low-moisture mozzarella can keep refrigerated for up to a month, though some pre-shredded low-moisture mozzarella is sold with a shelf life of up to 6 months.
Mozzarella is traditionally produced solely from the milk of the domestic water buffalo. After curdling the product is drained and the whey discarded. The cheese is then stretched and kneaded to produce a delicate consistency -- this process is generally known as pasta filata. According to the Mozzarella di Bufala trade association, "The cheese-maker kneads it with his hands, like a baker making bread, until he obtains a smooth, shiny paste, a strand of which he pulls out and lops off, forming the individual mozzarella."  It is then typically formed into ball shapes or in plait. In Italy, a "rubbery" consistency is generally considered not satisfactory; the cheese is expected to be softer.