Boil the potatoes in their skins until tender. Let them cool slightly, then peel and mash them using a potato ricer or masher until completely smooth.
Combine the mashed potatoes with the flour, egg yolk, and salt. Gently knead until a soft, non-sticky dough forms. Be careful not to overwork the dough, or the gnocchi will be tough.
Divide the dough into four portions. Roll each portion into a long rope about 2 cm in diameter. Use a sharp knife to cut the ropes into 2 cm pieces.
You can shape the cut pieces with a fork to create a ridged surface, which will help the sauce cling to the gnocchi.
Bring a large pot of salted water to a boil. Drop the gnocchi into the boiling water and cook until they float to the surface (about 2-3 minutes). Drain well.
In a skillet, melt the butter over low heat. Add the finely chopped walnuts and toast gently until fragrant, being careful not to burn them.
Pour in the heavy cream and simmer, stirring constantly, until the sauce has slightly thickened.
Stir in the grated Parmesan cheese and continue to stir until it is completely melted and incorporated into the sauce.
Add the cooked gnocchi to the skillet and gently toss to coat them evenly in the walnut cream sauce.
Serve immediately, garnished with fresh chopped parsley and extra toasted walnuts for added crunch.
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