Preparation

Place whole potatoes in a large saucepan, and cover with water by 1 inch. Cover and bring to a simmer over high heat. Uncover, salt generously, and simmer until the potatoes are very tender, about 10 minutes, then drain. When cool enough to handle, peel the skins and press the potatoes through a ricer or mash very well. Let cool. (Can be prepared and refrigerated several days ahead.)

In a large bowl, combine the potatoes, flour, and salt, and toss to coat. Make a well in the center, and add the egg yolks. Stir until the eggs are blended, then mix into the potatoes until a dough forms. Using your hands, continue mixing until the dough feels cohesive, adding a little more flour as necessary so it’s not sticky.

Lightly flour a work surface. Pinch off a handful of dough, roll it in the flour to coat, then use both hands to roll it into a 1/2-inch-thick rope. Using a knife or bench scraper, cut the rope into 1-inch lengths. Roll each length between your lightly floured palms to create a 2-inch dumpling tapered at both ends. Repeat with the remaining dough.

Bring a large pot of generously salted water to a boil. Add the dumplings and cook until they float, about 1 minute. Cook 1 minute more, then drain.

Heat the butter in a large skillet over medium-high heat. Add the dumplings and cook, stirring occasionally, until browned on all sides, about 5 minutes.