Preheat the oven to 325 degrees and line two sheet pans with parchment paper. In the bowl of a food processor, add the walnuts. Pulse until coarsely ground (about 30 short pulses).
Remove 1/2 a cup of the walnuts and set aside in a medium-sized bowl.
Continue processing the remaining walnuts until finely ground and clumping together slightly. You should see a little of the walnuts' oils gathering on the side of the food processor bowl.
Add the finely ground walnuts to the coarsely ground walnuts in the bowl. To the nuts, add the flour and salt. Toss to combine.
Without washing the food processor, add the sugar. Process until finely ground—about 30-40 seconds. Set aside.
In the bowl of a stand mixer, cream the butter on medium-low speed until smooth.
Add in the finely processed sugar and the vanilla. Beat until fully incorporated.
Turn off the mixer and add in the flour nut mixture to the butter and sugar.
Stir until all ingredients are fully combined and no streaks of flour are showing. You can do this with a stand mixer or by hand. The dough will be stiff.
Scoop out the cookie dough 1 tablespoon at a time and roll into a ball.
Place the cookies on a parchment paper-lined sheet pan. Leave about an inch of space between each cookie, they don't spread much. Bake at 325 degrees for 20-23 minutes until the bottom edges have lightly browned.
Remove the cookies from the oven and allow them to cool for 20 minutes.
In a shallow dish, add the powdered sugar. Toss the slightly warm cookies in the powdered sugar and gently roll around the dish to coat.
Set the coated cookies aside on a cooling rack or back on the cookie sheet to finish cooling.
Once all the cookies are completely cooled, give one final toss in the powdered sugar to fully coat and serve.