In the large mixing bowl, combine the butter, sugar, salt, and vanilla. Using the electric mixer set on medium speed, beat the mixture until it is smooth. Turn off the mixer a few times so you can scrape down the sides of the bowl with the rubber spatula. Add the flour and nuts. Continue mixing until the dough looks like moist pebbles, then turn off the mixer.
Gently squeeze the dough together with your hands. When the dough comes together in a mass, place it on a large piece of plastic wrap. Using your hands, shape it into a log 10 inches long and about 2 inches in diameter. Wrap the log in the plastic wrap and refrigerate until firm, about 3 hours.
Preheat the oven to 350 degrees F. Line the 3 baking sheets with aluminum foil.
Unwrap the dough and place it on the cutting board. Using the knife, cut the log into slices ¼ inch thick. Arrange the slices about 1 inch apart on the foil-lined baking sheets. Place 1 baking sheet in the oven and bake the cookies until their edges are golden brown, 14 to16 minutes. Using oven mitts, remove the baking sheet from the oven and set on a rack to cool completely. Repeat with the second and then the third baking sheets.
Lift the cooled cookies off the baking sheets with your fingers. Store the cookies in an airtight container. Makes about 40 cookies