Preheat oven to 400°F. Line a baking sheet with foil and place potatoes on it. Prick potatoes all over with a fork, cover tightly with foil, and bake for 45 minutes to 1 hour, or until cooked through and soft. Set aside until cool enough to handle.
Peel potatoes, cut into chunks, and pass through a potato ricer fitted with a fine disc. Stir in 4 tablespoons butter, parmesan and egg yolks; season with salt and pepper.
Line 2 baking sheets with parchment paper. Transfer half the mixture to a large piping bag fitted with a large star tip. Pipe potatoes onto baking sheets into 3-inch wide swirls. Freeze until firm, about 20 minutes. Meanwhile, increase oven to 450°F.
Bake until golden brown on the edges, 20-25 minutes. Brush with remaining 2 tablespoons melted butter. Serve hot.