Using an electric mixer, beat the butter in a large bowl until light and fluffy. Add the sugar and beat until well blended. Beat in the flour, and 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.
Preheat the oven to 325 degrees F.
Working with half of the chilled dough at a time while 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 2 days ahead. Store in an airtight container at room temperature). Sift additional powdered sugar over the cookies, if desired, before serving.