1. Beat together butter and 1 cup confectioners sugar in bowl until smooth and creamy. Add the vanilla.
2. On low speed, beat in flour and nuts. Wrap dough in plastic wrap; refrigerate until firm, 1 to 2 hours.
3. Heat oven to 325 degrees F. Pinch off pieces of dough in rounded teaspoonfuls. Roll into logs. Taper ends; bend into crescents. Place on ungreased baking sheets.
4. Bake in 325 degree oven for 19 to 20 minutes or until lightly browned.
5. Remove cookies to wire rack. Sprinkle with heavy layer of confectioners sugar. Cool completely. Sprinkle again with confectioners sugar. Makes 4-1/2 dozen.