This is the Tuscan version of grilled cheese, which my dad often made for me when I was a kid. There's real pleasure in an eggy sandwich fried in a way that seals in melted mozzarella. This warm, rustic panino is so beloved where I'm from that you can find it on most restaurant menus in Florence, plated as if it were no less meal-worthy than a bowl of pasta or meat dish.
Lay one-fourth of the mozzarella between 2 slices of bread and close. Repeat with the remaining bread slices.
In a small shallow baking dish, whisk together the eggs and milk. Season with salt and pepper to taste.
Meanwhile, in a large nonstick skillet, heat the olive oil over medium-high heat until hot. Dust 1 sandwich with flour, dip it in the egg mixture, then add it to the pan. Fry about 3 minutes on each side, until the outside is golden and crisp, but not burned, and the mozzarella is completely melted. Remove the sandwich from the pan, then repeat with the other 3 sandwiches. Cut each sandwich in half and serve.