Place your peeled and cubed potatoes into a large pot and cover them with cold water. Add a generous pinch of salt and bring the water to a boil over high heat. Reduce the heat to a simmer and cook 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 over low heat, shaking the pan gently. This helps evaporate any excess moisture, which is the secret to a potato swirl that holds its shape during baking.
Mash the potatoes using a potato masher or, for the smoothest results, pass them through a potato ricer into a large bowl. While the potatoes are still hot, stir in the softened butter, heavy cream, salt, pepper, and nutmeg. Mix until everything is completely incorporated and the mixture is silky.
Let the potato mixture cool for a few minutes so the eggs do not scramble upon contact. Once slightly cooled, add the egg yolks and stir vigorously until the mixture is uniform and holds its structure. The egg yolks are essential for providing the rich color and helping the potatoes set in the oven.
Preheat your oven to 400 degrees Fahrenheit and line a 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 mounds, swirling upward to create a decorative point. Space them about 2 inches apart on the tray.
Gently brush the outside of each potato mound with a little bit of melted butter. Place the baking sheet in the oven and bake for 15 to 20 minutes. You want the edges to be a beautiful golden brown and the peaks to be slightly toasted. Serve them immediately while they are hot and crisp.
