There is no clear agreement on what the actual original recipe was for Delmonico Potatoes.
For many decades, the practice has been to make them as follows:
Just slightly-underboiled potatoes are grated into long strands, then simmered for a short time with some milk, parmesan and seasonings. The mixture is then put into an oven-proof dish, sprinkled with grated parmesan, and baked until the potato is cooked through, the mixture is piping hot and the top is golden brown.
Toward the end of the 1900s, simplified versions became popular. Some of these versions just used mashed potato. Others used diced potato; others just swapped in a white sauce for the milk the potato was simmered in.
One version now even contains rice. Though the origin of this version is unknown, and no sources are given, it comes with a claim that it was in someone's family for "over 100 years", and was originally procured from the famous Delmonico chef, Charles Ranhofer.
Some versions add buttered bread crumbs as a topping. This is not particularly new, though: Fannie Merritt Farmer's version did in the 1890s.
On the menu of the revived Delmonico's today, they are referred to as "Delmonico's Baked Cream Potatoes."
Some think Delmonico Potatoes first appeared on the menu at Delmonico's in 1838 under the name of "Pommes de terre a la maitre d'hotel", meaning roughly "specialty potatoes of the house." On the menu, though, they gave the English translation as "fricasseed potatoes."
Some think this is what we now know as "Delmonico Potatoes", and feel that the English translation is puzzling given that it is actually a baked, gratin dish.
"Cut the potatoes with a vegetable cutter into small balls about the size of a marble; put them into a stew-pan with plenty of butter, and a good sprinkling of salt; keep the sauce-pan covered, and shake occasionally until they are quite done, which will be in about an hour." 
Alessandro Filippini, a former chef at Delmonico's, published a book in 1906 called "The International Cook Book." On page 204, he gave his recipe for Delmonico potatoes as:
"[Recipe Number] 718. Delmonico Potatoes
Place four good-sized boiled and finely handshed [Ed: hand-shredded] potatoes in a frying pan with one and a half gills [Ed: 6 oz] cold milk, half gill [Ed: 2 oz] cream, two saltspoons [Ed. 2 x 1/4 teaspoon] salt, one saltspoon white pepper, and a saltspoon grated nutmeg; mix well and cook on the range for ten minutes, lightly mixing occasionally. Then add one tablespoon grated Parmesan cheese, lightly mix again. Transfer the potatoes into a gratin dish, sprinkle another light tablespoon grated Parmesan cheese over and set in the oven to bake for six minutes, or until they have obtained a good golden colour; remove and serve."
 Fanny Lemira Gillette. White House Cook Book. Chicago: R.S. Peale & Co., 1887. page 174.
Literature & Lore
To Potatoes an Gratin add one-third cup grated mild cheese, arranging potatoes and cheese in alternate layers before covering with crumbs.
-- Fannie Merritt Farmer. The Boston Cooking-School Cookbook . Boston, Little, Brown And Company (1896), page 285.
Potato DishesAlabaster; Anna Potatoes; Berny Potato Croquettes; Champ; Delmonico Potatoes; Duchess Potatoes; French Fries; Hash Browns; Hash; Lyonnaise Potatoes; Mashed Potato; Matchstick Potatoes; Papas Arrugadas; Pâté Râpé; Potato Dishes; Potato Haystack; Potato Korv; Potato Pancakes; Rappie Pie; Rumplethumps; Salt Potatoes; Stovies; Vichyssoise
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-- Ludwig Bemelmans (American author. 27 April 1898 - 1 October 1962)