Around the world, these buttery, nutty cookies go by lots of different names: Mexican Wedding Cookies, Russian Tea Cakes, Sand Tarts, Polvorones, and more.
Adjust oven rack to middle position and heat oven to 325 degrees. Line rimmed baking sheet with parchment paper.
Place walnuts in large zipper-lock plastic bag and seal, removing as much air as possible from bag. Use rolling pin to gently pound walnuts into very small pieces.
Even though this photo shows peanuts, the technique is the same!
In medium bowl, whisk together pounded walnuts, flour, and salt.
Add softened butter and sugar to bowl of stand mixer (or large bowl if using handheld mixer). If using stand mixer, lock bowl into place and attach paddle to stand mixer. Start mixer on medium-high speed and beat until pale and fluffy, 3 to 4 minutes. Stop mixer and use rubber spatula to scrape down sides of bowl.
Add vanilla and flour mixture. Start mixer on low speed and mix until combined, about 45 seconds. Stop mixer. Remove bowl from stand mixer, if using. Use rubber spatula to scrape down sides of bowl and stir in any remaining dry flour.
Use your hands to roll dough into 24 balls (about 1 tablespoon each). Place dough balls on parchment-lined baking sheet, leaving space between balls.
Place baking sheet in oven. Bake cookies until tops are pale golden and bottoms are just beginning to brown, about 18 minutes.