Thomas Keller's Black Walnut Snow Cookies

Preheat oven to 350 degrees. Using a stand mixer with the paddle attachment, beat butter until light and fluffy. Add the 2 3/4 cups powdered sugar, salt, and vanilla and mix until well blended. Add the flour and then the walnuts to form a dough. Divide the dough in half and form each half into a ball. Wrap each ball in plastic wrap and refrigerate until cold, about 40 minutes. Working with half of the chilled dough at a time, use a small ice cream scoop or your hands to roll into balls about 2 teaspoons large. Arrange the balls on baking sheets, spacing them 1/2 inch apart. Bake cookies until they are golden brown on the bottom and just pale golden on top, about 18 minutes. Cool for 5 minutes on the baking sheets, then gently toss warm cookies in the powdered sugar to coat completely. Transfer coated cookies to a cooling rack to cool. Sift remaining powdered sugar over cookies and serve.