Step 1Mix the eggs, milk, salt, pepper and Parmesan in a bowl until well blended. Set aside.
Step 2Grease an 8-inch square baking dish with 1 tablespoon of the butter.
Step 3Place 4 of the bread slices in the bottom of the baking dish. Sprinkle the Fontina cheese evenly over the bread. Place the remaining slices of bread on top. Pour the custard batter over the sandwiches. Cover with wax paper or plastic wrap and let stand for at least 15 minutes.
Step 4Heat the oven to 350 degrees.
Step 5Bake uncovered until the sandwiches are puffed and lightly golden, about 25 to 30 minutes. Remove the dish from the oven and keep warm.
Step 6Melt the remaining 2 tablespoons of butter in a large skillet over medium heat. Remove the sandwiches from the baking dish with a spatula and place in the pan. Cook until golden brown, 2 to 3 minutes a side. Cut the sandwiches in half and serve hot.