In a bowl, with an electric mixer on medium speed, or in a food processor, beat or whirl butter, 1/3 cup powdered sugar, and the vanilla and flour until smooth and well blended. Stir in pecans.
Shape dough into 1 1/4-inch balls and place about 1 inch apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets.
Bake in a 300° oven until cookies are pale golden brown, 20 to 25 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. Let cool on sheets for 5 minutes.
In a shallow bowl, mix remaining 1 1/3 cups powdered sugar with cardamom. Gently turn warm cookies in powdered sugar mixture, a few at a time, then set on racks to cool. When cool, coat cookies again with powdered sugar mixture.