Scrub potatoes in cold water. Boil for between an hour and an hour and a half, depending on the size of the potatoes. Here's how to tell when the potatoes are cooked: Push the blade of a knife through the center of the potato while it's still in the pot. Pull the knife up. If the knife slides out, the potato is done. If it picks the potato up out of the water, it's not done.
When cooked, remove potatoes from the water. While still hot, peel and slice the potatoes, and run them through a food mill to mash smooth, with no lumps. (I find that this cannot be done properly in a food processor or blender.)
Melt the butter in a deep skillet or saucepan over low heat. Add the mashed potatoes and stir vigorously to mix. This will create a very stiff mixture, almost like bread dough. Stir egg yolks, one at a time, into the potato mixture. Do this quickly, before yolks have a chance to cook from the heat of the potatoes, and keep stirring. When eggs are incorporated completely into potatoes, remove from stove.
Scoop the potato "dough" on top of a clean, smooth surface dusted with flour. Add flour, about a half-cup at a time, and knead it into the potatoes, using hands and a plastic scraper. Sprinkle a little extra flour over the potato-dough and counter if necessary to prevent sticking.
Cut off about one-fifth of the dough ball, and roll it out to a long, thin (about a half-inch in diameter) "snake." With a flour-dusted knife, cut off pieces of the "snake" about a half-inch long.
With your thumb, roll each piece of dough along the tines of a table fork. As you do this, press down a little so the dough curls in on itself a little. This will result in a nugget with a large indentation on one side and ridges across the other--the classic gnocchi shape. Put the gnocchi on a floured pan in one well- spaced layer.
This recipe makes about 300 gnocchi--enough for six large entrees or 12 appetizers. But you can preserve a portion for later use. Allow the gnocchi to dry for about two hours on the pan, then put them into plastic bags and freeze. When you're ready to use the frozen gnocchi, put them right into the boiling water--do not defrost first.
Cook the gnocchi exactly as you would pasta. Boil them for about four minutes (longer, of course, if frozen). Drain them and toss them in a skillet with tomato sauce, pesto sauce, or sage butter sauce.
