How to Make It
Preheat the oven to 400°. Grease a 1-quart gratin dish with butter and dust with the Parmigiano; refrigerate. In a saucepan, melt the 3 tablespoons of butter over moderate heat. Whisk in the flour and cook, whisking, for 1 minute. Whisk in the milk, bring to a boil and cook, whisking, until thickened, 1 minute. Remove the béchamel from the heat, then whisk in 4 egg yolks along with the salt and pepper; reserve the remaining yolk for another use.
In a clean bowl, beat the whites until firm peaks form. Whisk one-third of the whites into the béchamel, then fold in the remaining beaten whites. Fold in the Gruyère and chives; scrape into the prepared dish. Arrange the American cheese strips on top in a crisscross pattern. Bake for 25 minutes, until puffed and golden. Serve.