Put 1/2 cup of the walnuts and the salt in a food processor and pulse until finely ground. Roughly chop the remaining 1 cup walnuts.
Position two oven racks in the upper and lower thirds of the oven and preheat to 325 degrees F.
Beat the butter in a large bowl with an electric mixer on medium-high speed until smooth, 2 to 3 minutes. Add the granulated sugar and beat until light and fluffy, about 2 minutes. Scrape down the sides of the bowl. Beat in the flour, then the ground and chopped walnuts. Divide the dough in half, forming each half into a ball. Wrap separately in plastic and chill until cold, about 30 minutes.
Put the confectioners' sugar in a large bowl.
Working with half of the chilled dough at a time and keeping the rest in the fridge, roll the dough by 2 teaspoonfuls between your palms into balls. Arrange the balls on a large baking sheet, spacing them 1/2 inch apart.
Bake the cookies until golden brown on the bottom and just pale golden on top, about 18 minutes. Cool the cookies for 5 minutes on the baking sheet. Toss the warm cookies in the powdered sugar. Transfer the sugar-coated cookies to a rack to cool completely. The cookies can be prepared up to 2 days ahead. Once cooled, store them in an airtight container. You need to make sure they are cooled before storing them, otherwise they will get soggy. Sift additional powdered sugar and cinnamon over the cookies if desired before serving.