Preheat oven to 325 degrees F with racks in upper- and middle-third positions.
Pulse walnuts in the bowl of a food processor or blender until very fine (as Heatter notes, it's ok for the walnuts to develop an oily appearance as they're ground). Set aside.
In a bowl of a stand mixer fitted with the paddle attachment, cream butter on medium speed until smooth. Add the vanilla and ground walnuts and mix until combined, stopping and scraping the bowl as needed. Add the granulated sugar, and beat until incorporated.
With the mixer on low speed, add the sifted flour, 1/2 cup at a time, until combined, scraping the bowl once or twice.
Place a piece of wax paper on your countertop. Using a one teaspoon measuring spoon, divide the dough into rounded portions. Roll each ball between your cupped palms until about 2-1/2 inches in length. The roll should be slightly thicker in the middle, with tapered ends.
Curve the rolled dough into a "C" shape and place on parchment-lined baking sheets, about 1-inch apart.*
Bake cookies 16-18 minutes, rotating pans halfway through. The finished cookies should be lightly-browned on the tapered ends, and still light in the center. Remove cookies from oven and let stand 1-2 minutes.
When cookies have slightly firmed, use a thin metal spatula to gently transfer them to wire cooling racks set over wax paper. The cookies are very delicate, so don't be surprised if a few break.
While the cookies are warm, dust generously with confectioner's sugar. When cooled to room temperature, store in several layers between pieces of wax paper in an airtight container for up to a week. If cookies are stored, re-dust with powdered sugar before serving for the prettiest presentation.