Place the peeled and cubed potatoes in a large pot and cover them with cold water. Add a pinch of salt and bring the water to a boil over high heat. Reduce the heat to medium and simmer for about 15 to 20 minutes, or until the potatoes are very tender when pierced with a fork.
Drain the potatoes thoroughly in a colander. To ensure the best texture, return the drained potatoes to the hot pot for a minute or two over low heat. Shake the pot gently to cook off any remaining moisture, which helps prevent the potato mounds from losing their shape in the oven.
Use a potato ricer or a fine masher to process the potatoes into a large bowl until they are completely smooth and free of lumps. While the potatoes are still hot, stir in the softened butter, heavy cream, salt, pepper, and nutmeg. If you are using Parmesan cheese, fold it in now until the mixture is well combined.
Allow the potato mixture to cool slightly for a few minutes so the eggs do not scramble upon contact. Add the egg yolks one at a time, stirring vigorously after each addition. The egg yolks are the secret ingredient that helps the potatoes hold their beautiful piped shape and adds a rich, golden color.
Preheat your oven to 400 degrees F and line a large baking sheet with parchment paper. Transfer the potato mixture into a large piping bag fitted with a large star tip. Pipe the potatoes into 3 inch wide swirls, starting from the center and moving outward and upward to create a decorative peak.
Carefully brush the potato mounds with a little bit of melted butter. Place the baking sheet in the oven and bake for 15 to 20 minutes, or until the edges and ridges of the potatoes are beautifully browned and crisp. Serve them immediately while they are hot and fresh.
