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.
In a large bowl, combine the mashed potatoes with the flour, egg yolk, and salt. Gently mix until a soft, non-sticky dough forms. Be careful not to overwork the dough, as this will result in tough gnocchi.
Divide the dough into four equal portions. On a lightly floured surface, roll each portion into a long rope about 2 cm in diameter. Use a sharp knife to cut the ropes into 2 cm pieces.
If desired, shape each gnocchi piece by gently rolling it over the tines of a fork to create ridges, which will help the sauce cling better.
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 dry skillet, toast the walnuts over low heat until golden brown and fragrant. Set aside.
In a skillet, melt the butter over low heat. Add the heavy cream and warm gently.
Crumble the goat cheese into the cream sauce and stir until it is completely melted and smooth.
Stir in the grated Parmesan cheese until melted and smooth.
Add the cooked gnocchi to the skillet and gently toss to coat them evenly with the goat cheese sauce.
Serve immediately, garnished with fresh chopped parsley, toasted walnuts, and extra Parmesan cheese.
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