Heat the oven to 400 degrees. Poke the potatoes a few times with a fork and place directly on the oven rack. Bake about 1 hour and 20 minutes, or until tender when pierced.
When the potatoes are cool enough to handle, remove the skins (or scoop the flesh out with a spoon). Push them through the large holes of a food mill or ricer into a large bowl, or mash them by hand with a potato masher or fork.
Add the salt, egg yolks and 2 cups flour and mix together to form a shaggy dough. Turn the mixture out onto a floured surface and knead, adding a little more flour if needed, until the dough is soft and smooth.
Divide the dough into portions the size of the palm of your hand. On a floured surface, roll each into a rope about 1/2-inch thick. Cut into 1-inch lengths. Roll each wad of dough firmly over a gnocchi board or the concave side of a fork. Arrange the gnocchi on a floured baking sheet as you go.
Bring a large stockpot of salted water to a boil. Drop the gnocchi, in batches, into the water and boil until they bob to the surface, about 3 minutes. Scoop out the cooked gnocchi into a colander set over a bowl to drain. Scoop 1/4 cup of the cooking water and reserve.
To serve, stir a about a tablespoon of reserved cooking water into the pesto sauce until it's smooth. Toss the gnocchi gently with the sauce in a serving bowl, and top with grated Parmesan.
To make the pesto:
Put the walnuts in a food processor and pulse until finely chopped. Add the parsley, cheese, olive oil, salt and garlic.
Process until the pesto is smooth.
Freezing potato gnocchi:To be sure you always have a late-night gnocchi stash on hand, make a batch of gnocchi and freeze them, Lay them out in one layer on a tray and place in the freezer.When they're frozen solid, transfer the gnocchi to zippered bags. Keep frozen up to 3 months before cooking.