In a stand mixer, or with a handheld mixer, cream the butter until light.
To the bowl, add the powdered sugar, salt, and vanilla. Mix with the butter until light, about 3-4 minutes, scraping the sides of the bowl as necessary.
Add the flour and chopped walnuts. Mix on low until combined, scraping the sides of the bowl as necessary.
Place the dough in wax paper and chill in the refrigerator for 1 hour.
Take the dough out of the refrigerator and divide into 8 sections. Each section should make about 5 cookies.
Heat the oven to 350 degrees F.
To shape the crescent cookies, take a little dough from each section and roll into a small ball between your hands. Then roll into a sausage shape, trying to taper at the ends. Form with your hands into a crescent shape, then place on a cookie sheet to bake.
Bake for 12-16 minutes. Cookies should feel slightly firm (but will firm up more as cooling), but not browned.
Let cool on the cookie sheet, and then transfer to a cooling rack or a plate.
Dust with powdered sugar.