Cream the butter thoroughly until there are no lumps and it is fluffy, in a stand mixer fitted with a paddle attachment or a medium mixing bowl and wide wooden spoon or sturdy spatula.
Beat in the powdered sugar. Sift it first if it is very lumpy, especially if you are mixing by hand. Cream the sugar into the butter until they are smooth and evenly mixed.
Add the ground walnuts, salt, and vanilla to the butter and sugar mixture, whipping them together until well incorporated.
Add in the flour and mix just until all ingredients are combined into a stiff dough. You can shape the cookies immediately, or cover the dough well and chill it for up to three days.
Preheat the oven to 350 ℉, and either grease your cookie sheets, or line them with nonstick silicone baking mats or parchment paper.
Shape the crescents. Pinch off walnut-sized balls of dough, rolling them into rough balls, then squeezing or rolling them between your palms to make a cylinder. Roll the ends of the cylinders into tapered points, then bend gently into crescent shapes. If they crack in the center, press the dough back together firmly.
Space them out on the cookie sheets, allowing a little bit of space for them to spread. If the dough has warmed significantly while you are shaping it, you can chill the trays before baking them to help them keep their shape. You can always bake a test cookie to be sure.
Bake the cookies 14-16 minutes, or just until they are set. They will still be soft until they cool. Remove them when they just begin to color around the edges. Let them cool on racks, or on the sheets if they aren't close to overbaking.
Once the cookies are cooled, dust them with the additional powdered sugar using a wire sieve or sifter. Store in airtight tins at room temperature for up to a week, or freeze for several months.
Notes
Note on nutritional information: Calculations are an estimate, using chopped walnuts instead of ground walnuts, and for a yield of 25 crescent shortbread cookies. Exact values may vary slightly with cookie size, and using more or less powdered sugar for garnish.