Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Spread the walnuts on a rimmed baking sheet. Bake on the lower rack until toasted, 10 to 12 minutes; let cool completely. Pulse the nuts in a food processor until finely ground. Add the flour and pulse to combine; set aside.
Beat the butter in a large bowl with a mixer on medium-high speed until light and fluffy, 2 to 4 minutes. Add 1/2 cup confectioners' sugar, the vanilla, cardamom and salt; beat until combined. Reduce the mixer speed to low and beat in the walnut-flour mixture until just incorporated. Cover the dough and refrigerate until slightly firm, about 30 minutes.
Place the remaining 1 cup confectioners' sugar in a shallow bowl; set aside. Shape tablespoonfuls of dough into thin crescents and arrange 1 inch apart on 2 unlined baking sheets.
Bake, switching the pans halfway through, until the cookies are golden on the bottom, 20 to 25 minutes. Let cool 5 minutes on the pans, then carefully transfer to the bowl of confectioners? sugar and gently roll to coat. Transfer to a rack to cool completely, then reroll in the confectioners? sugar.