Preheat oven to 400°. Wash potatoes and pierce them all over with a fork. Bake for 1 hour, until soft. Alternatively, bake the potatoes in a microwave oven for 15–20 minutes, turning over every 5 minutes, until soft.

Cut the potatoes in half lengthwise and scoop the flesh onto work surface. Using a sieve or fine strainer, press the potatoes through to create a fine texture. Transfer potatoes to a bowl. Stir in egg yolks, flour and Parmesan. Knead until a stiff-smooth dough forms. The dough should be an even texture and slightly tacky, not sticky.

Line a baking sheet with a dusting of flour. Flour work surface and cut the dough into 4 pieces, rolling each into a ¾-inch-thick rope shape. Cut the ropes into ¾-inch pieces. Roll each piece against the tines of a fork (or a gnocchi board) to make ridges; transfer to the baking sheet.

In a large pot of salted, simmering water, cook the gnocchi until they rise to the surface, then simmer for 1 minute longer. In a large nonstick skillet on medium-high heat, melt some butter. Using a slotted spoon, add the gnocchi to the butter and cook until golden brown. Sprinkle with more Parmesan cheese and serve.

To make ahead: The uncooked, formed gnocchi pieces can be frozen on the prepared baking sheet, then transferred to a resealable plastic bag and frozen for up to 1 month. Boil without defrosting.