1. Position rack in center of the oven and preheat to 375 degrees F.
2. In a large bowl combine the butter with 1/2 cup of the confectioner's sugar; beat with an electric mixer until fluffy, about 1 minute. With a spoon, stir in the vanilla, flour, salt and walnuts to form a stiff dough.
3. Using about 2 teaspoons of dough for each cookie, shape half the dough by rolling between hands. Place about 1 1/2 inches apart on an ungreased baking sheet and bake about 12 minutes, until just pale golden on the bottom.
4. Meanwhile, sift 1/2 cup of the remaining confectioner's 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, tossing them around to coat; let the cookies cool completely in the sugar and then sift 1/4 cup more sugar over them to coat.
5. Repeat with the remaining cookie dough, using a cool baking sheet to bake them and the remaining 3/4 cup confectioner's sugar, divided as before, to coat. Makes 4 to 5 dozen