In the bowl of a stand mixer fitted with paddle attachment, cream the butter and vanilla together until smooth.
In a separate bowl, combine the 6 Tbsp. confectioners' sugar with the flour. Add to the butter mixture and stir just until everything is well combined. Stir in the finely chopped nuts.
Roll dough into 1-inch balls and set on an ungreased baking sheet about 2 inches apart. Bake for about 12 minutes until cookies are set and the bottoms are just beginning to turn golden-brown.
While still warm, gently roll the cookies in confectioners' sugar and set on a piece of parchment paper or waxed paper to cool. Once the cookies have cooled completely, roll in confectioners' sugar a second time.