Preheat the broiler, setting the rack 6 inches from the heat. Arrange the bread on a work surface and spread 1 side only with the mayonnaise, evenly and to the edges. Sprinkle the Parmesan over the mayonnaise and press lightly to make it adhere. Flip the 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.

From The Portlandia Cookbook: Cook Like A Local