Preheat the oven to 350 degrees F (175 degrees C). Arrange a rack in the center position of the oven. Line a rimmed baking sheet with a silicone mat.

Place butter, walnuts, 1/3 cup packed powdered sugar, vanilla, and salt in a large bowl. Top with 2 cups plus 2 tablespoons flour. Mix with clean hands until dough starts to clump up. Keep mixing by hand until all flour and clumps of butter are evenly mixed into dough.

Scoop out dough and roll by hand into uniformly round balls, just slightly larger than 1 inch. Place about 2 inches apart on the prepared baking sheet.

Bake in the preheated oven until lightly golden, 15 to 25 minutes depending on the size of cookies.

Let cool for exactly 5 minutes, then roll warm cookies in 1 cup confectioners’ sugar in a medium bowl. Let cookies cool completely, then toss them again in confectioners’ sugar.