Preheat oven to 375F (190C) and position rack in the center.

In a large bowl, combine the softened butter with 1/2 cup of the sifted powdered sugar.

Beat with an electric mixer until fluffy, about 1 minute.

Stir in the vanilla extract, all-purpose flour, salt, and chopped walnuts to form a stiff dough.

Shape half the dough into small balls using about 2 teaspoons of dough for each cookie.

Place the shaped dough about 1 1/2 inches apart on an ungreased baking sheet.

Bake for about 12 minutes, until just pale golden on the bottom.

While the cookies are baking, sift 1/2 cup of the remaining powdered sugar into a 13 by 9 inch pan.

As soon as cookies are removed from the oven, use a spatula to transfer them to the sugar.

Toss them around to coat completely.

Let the cookies cool completely in the sugar.

Sift 1/4 cup more sugar over them to coat again.

Repeat with the remaining cookie dough, using fresh sugar, divided as before, to coat the cookies after baking.