Add potatoes to a large pot of very well salted cold water and bring to a boil over high heat. Reduce heat to medium-low and simmer until potatoes are very tender but not falling apart, 30 to 45 minutes. Drain very well, and let sit in strainer for at least 5 minutes.

Transfer to a bowl, and mash potatoes until smooth. Add 3 tablespoons butter, salt, cayenne, and nutmeg, and continue mashing and mixing until butter completely disappears and mixture is smooth.

Add cream and egg yolks, and whisk vigorously until mixture is very smooth. Taste to check seasoning and adjust to your liking. Cover and let mashed potatoes cool to room temperature.

Preheat the oven to 450 degrees F (220 degrees C). Line a baking sheet with parchment paper.

Spoon mixture into a piping bag. Pipe 12 portions on the prepared baking sheet (alternatively you can scoop out 12 portions). Chill potatoes in the refrigerator until you're ready to bake them.

Drizzle or lightly brush melted butter over each duchess potato.

Bake in the preheated oven until well browned, about 20 minutes.