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.