This dish has many different names, but no matter what you call it, it is delicious. It makes a great breakfast, brunch or lunch, but you can serve it for dinner as well!
4 slices best-quality packaged white sandwich bread
2 T. mayonnaise (not reduced-fat)
2 T. freshly grated Parmigiano-Reggiano
3 oz. (about 1 cup) coarsely shredded cheese, such as Fontina, unaged Asiago or unaged Gouda
1 scallion, thinly sliced
unsalted butter, for the pan
2 large eggs
Truffle salt (optional)
Preheat the broiler, setting the rack 6 inches from the heat.
Arrange bread on a work surface and spread each 1 side only with the mayonnaise, evenly and to the edges. Sprinkle Parmesan over the mayonnaise and press lightly to make it adhere.
Flip bread and add the coarsely shredded cheese and the scallions to 2 of the slices. Close the sandwiches, with the Parmesan-encrusted sides out.
Heat an ovenproof griddle or skillet over medium heat and lightly grease with butter. Add the sandwiches and cook over moderate heat until lightly browned on the bottom, 1 to 1 1/2 minutes.
Transfer the sandwiches to a cutting board. Using a biscuit cutter or the rim of a glass, cut out a 2-inch hole in the center of each sandwich. Return the sandwiches and cutouts to the griddle, browned side up, and melt a sliver of butter in the holes.
Crack an egg into each hole and cook over medium heat for about 30 seconds. Transfer the griddle to the broiler and cook until the top is browned and the egg whites are just set but the yolk is still runny, about 2 minutes. The tops and bottoms should be perfectly golden, but be careful not to overtoast the bread.
Sprinkle the eggs with truffle salt, if using, and serve immediately, with the cutouts of bread on the side.
Makes 2 sandwiches