Toast the walnuts in a large skillet over medium heat until they are golden and smell nutty, 2 to 4 minutes. Transfer most of the walnuts to a food processor and grind until fine, then pulse the remaining walnuts separately until they are coarsely chopped to provide texture.

Simmer the wine in the empty skillet over medium-high heat until it has reduced significantly and looks like a thin syrup, about 3 minutes. Whisk in the cream, ground and chopped walnuts, Parmesan, salt, and pepper, then remove from the heat and cover to keep the sauce warm.

Bring 4 quarts of water to a boil in a large pot. Add the salt and pasta and cook until tender but still al dente, which should take 2 to 3 minutes for fresh noodles; reserve some of the cooking water before draining the pasta.

Combine the pasta and a splash of the cooking water with the sauce in the skillet and cook over medium heat, tossing constantly until the noodles are well coated and steaming. Adjust the consistency with more cooking water if the sauce looks too thick, then stir in the chives and serve immediately while the sauce is glossy and hot.