Combine the egg and milk in a large bowl, and beat lightly. Cut the French bread into four thick slices, and soak each piece in the egg mixture for around 30 seconds (making sure both sides of the bread are well covered). Sandwich each slice of fresh mozzarella in between two pieces of eggy bread, creating two sandwiches in total.
Melt the butter in a frying pan, and add the leek and garlic, along with some salt and pepper. Make a space in the middle of the pan, and add the mozzarella sandwiches. Cook over a fairly low heat for about 4 minutes, stirring the leeks every now and then, until the underside of the bread is lightly browned. Carefully turn the sandwiches over with a spatula, and repeat with the other side.
Sprinkle half of the parmesan onto the top of the sandwiches, and carefully turn them over once more so that the parmesan is underneath them (don’t worry if you make a it of a mess in the pan!). Cook for another 4 minutes, until the cheese underneath is crispy and the mozzarella in the centre has begun to melt. Add the remaining parmesan on top, flip the sandwiches once more, and cook until the cheesy bread is crispy on the outside, and the centres are fully cooked and melty.
Serve topped with the garlicky leeks.