In large bowl, w/ mixer on medium speed, beat butter, granulated sugar, and vanilla until creamy, occasionally scraping bowl with rubber spatula. Reduce speed to low; gradually beat in flour and salt just until blended, occasionally scraping bowl. Stir in walnuts.
Shape dough by rounded measuring teaspoons into 1-inch balls. Place balls, 1 inch apart, on ungreased large cookie sheet. Bake cookies 13-15 minutes or until bottoms are lightly browned.
Place confectioners' sugar in pie plate. While cookies are hot, with metal spatula, transfer 4 or 5 cookies at a time to pie plate with confectioners' sugar. gently turn cookies w/ fork to generously coat with sugar. Transfer cookies to wire rack to cool completely.
Repeat with remaining dough and confectioners' sugar. Store cookies in tightly covered container at room temperature up to 1 week or in freezer up to 3 months. Dust with additional confectioners' sugar before serving if you like.