In a large bowl, cream the softened butter and powdered sugar until light and fluffy—about 2 minutes. Stir in the vanilla extract.
Mix in the flour and salt until just combined. Fold in the finely chopped nuts. The dough will be soft but slightly crumbly and should hold together when pressed.
Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper. Scoop tablespoon-sized portions of dough and roll them into balls. Place about 1 inch apart on the baking sheet. Bake for 12–14 minutes, or until the bottoms are lightly golden and the tops remain pale.
Let the cookies cool for 5 minutes, then roll them in powdered sugar while still warm. Set aside on a rack to cool completely. Once cooled, roll again in powdered sugar for an extra snowy finish.