Add the butter, sugar, vanilla and salt into a large bowl and beat using an electric beater on high speed for 1 to 2 minutes or smooth and creamy.Scrape down the sides and bottom of the bowl as needed.
Add the flour mixture and chopped walnuts, mixing only until incorporated.
Divide the dough in half and then roll each half between two sheets of parchment until it is less than ¼ inch (.5 cm) thick. Then place dough on a baking sheet (along with the parchment paper) and chill in the refrigerator until firm, about one hour.
Once chilled, remove from refrigerator and peel off the top piece of parchment paper. Using a 2 ½ inch round fluted cookie cutter, cut out the cookies.
Place on parchment lined baking sheets, spacing the cookies about 2 inches (5 cm) apart. Put the baking sheet, with the unbaked cookies, in the refrigerator, while you preheat your oven to 350 degrees F (180 degrees C).
Meanwhile, gather up any scraps of leftover dough and re-roll. (You may need to refrigerate the dough again before cutting out the cookies.
Bake the cookies, one sheet pan at a time, for approximately 15 to 19 minutes or until they are a deep golden brown color. This ensures a crisp and crumbly cookie. Remove cookies from oven and cool completely on a wire rack.
Notes
Bake one sheet pan at a time. Chop raw walnuts into small pieces.