In a bowl, with an electric mixer on low speed, or in a food processor, beat or whirl butter, 1/3 cup powdered sugar, and the vanilla and flour until smooth and well blended. Stir in walnuts.
Shape dough into 3/4-inch balls and place about 1 inch apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets.
Bake in a 300 oven until cookies are pale golden brown, 15 to 20 minutes; if baking more than one pan at a time, switch pan positions halfway through baking.
Let cool on sheets for 5 minutes.
Place remaining 1 1/3 cups powdered sugar in a shallow bowl. Gently turn warm cookies in sugar, a few at a time, then set on racks to cool. When cool, coat cookies again with powdered sugar.
