1¼ cups walnut halves
1½ cups all-purpose flour
¼ teaspoon salt
1½ sticks (¾ cup) unsalted butter, room temperature
1½ cups confectioners' sugar
1 teaspoon pure vanilla extract
1. Preheat the oven to 350°F. Spread walnuts in a single layer on a rimmed baking sheet, and toast until light golden and fragrant, about 8 minutes. Transfer to a plate, and let cool completely. Line two baking sheets with parchment paper.
2. In a food processor, combine ½ cup flour with the toasted nuts, and pulse until the nuts are finely ground. In a medium bowl, whisk together the flour-nut mixture and the remaining 1 cup flour and the salt; set aside.
3. In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and ½ cup sugar until light and fluffy, about 2 minutes. Beat in vanilla. With the mixer on low speed, add the flour mixture in two batches, and beat until combined.
4. Roll 1 tablespoon of dough into a 3-inch log. Using your fingers, shape log into a crescent, tapering the ends slightly so that the center is the widest part. Repeat with remaining dough, placing crescents about 1 inch apart on prepared sheets. Freeze or refrigerate until crescents are very firm, about 30 minutes.
5. Bake, rotating sheets halfway through, until the edges of the cookies begin to turn golden, 16 to 18 minutes. Transfer sheets to a wire rack to cool, 5 minutes. Transfer cookies to the rack to cool completely. Place the remaining 1 cup sugar in a shallow bowl and roll cookies in it to coat completely. Cookies can be kept in an airtight container at room temperature for up to 5 days.