Peel the onion and, using a sharp stainless-steel knife, cut the onion into the thinnest-possible slices to make almost-transparent onion rings. There should be about ½cup of onion rings.
Heat the butter in a 4-cup saucepan and when butter is hot add the onion rings. Cover tightly and cook over low heat until onion is wilted. Sprinkle the onion rings with the flour and stir to distribute the flour evenly. Gradually stir in the milk, using a wire whisk. Bring to a boil, stirring, and when thickened and smooth add the cream and stir. Add salt and pepper to taste. Add the cayenne and nutmeg. Simmer for about 3 minutes.
Meanwhile, peel the eggs under cold running water and dry on paper toweling. Slice the eggs with an egg slicer or with a stainless-steel knife. Add eggs to the sauce and very gently stir them into the sauce with a two-pronged fork. Stir as little as possible and try not to break up the yolks. Heat thoroughly and serve on toast or with rice.