Cut the bread to roughly the same size as the mozzarella slices, allowing a little extra around the edges for the cheese to spread. Sandwich each slice of mozzarella between 2 slices of bread, pressing down lightly around the edges of each one to "seal." Whisk the eggs with the milk until fluffy, and season with salt and black pepper.
Heat enough oil for shallow-frying (depending on the thickness of your sandwiches) in a heavy frying pan over medium heat. Lightly coat each of the little sandwiches in flour, then dip in the beaten egg, allowing the bread to soak up the eggy mixture. Carefully lower the sandwiches into the very hot oil, cooking a few at a time.
Be careful not to crowd the pan, as the temperature of the oil will drop and your sandwiches will end up soggy, rather than crispy and light. Allow to cook for a few minutes until the cheese starts to melt and the bread is crisp and golden underneath. Using a spatula, turn the sandwiches over and press down. Lightly flatten with the spatula. Cook for another 2-3 minutes, then remove and drain on paper towels.
These sandwiches are best piping hot and fresh from the pan, so serve immediately - with a warning about the hot melted cheese inside.