Duchess potatoes are finely shredded, cooked potato, which is formed into shapes and baked, or used as a border on dishes.
The classical definition of how to make them is given by Larousse.
You peel the potatoes, and boil them in salted water. Drain the potatoes, and let them sit in the hot pan or in a hot oven for a few minutes for excess moisture to evaporate off.
Then you press them through (in order of preference) either a sieve, a potato ricer, or, mash them with a potato masher. Put them after this in a hot pan, and let them sit for a few more minutes for more moisture to evaporate.
Then, season the potato with salt, pepper and nutmeg, then stir in butter, plus 1 whole egg plus 2 egg yolks.
Spread out the potato mixture and let it sit till it gets cold.
Then, make it into shapes as required. For croquettes, the shapes can be formed by hand or piped into small mounds.
If the mixture is going to be piped through a piping bag (aka pastry bag) for use as borders, etc, then it is best used while still somewhat warm.
The potato mixture can also be baked whole as an entire dish.
Before baking in any form, it is usually brushed with beaten egg.
When baking in shapes or as a border, the mixture needs about 10 to 15 minutes to get golden brown, depending on the size of the shapes or border.
You can bake them, or brown them under a broiler (aka grill in the UK) or salamander.
Many versions now add milk or cream, but there is no milk or cream in the classical French version, nor is there any cheese.
Both additions sound lovely, but then the potatoes are something else, not Duchess Potatoes — perhaps Grand Duchess?
The mixture for Duchess Potatoes can be made up to 24 hours in advance and stored in a refrigerator, covered. For that matter, you can also make the shapes in advance and store in the same way.