Preheat your oven to 325 degrees F and line a baking sheet with parchment paper or a silpat mat. With a cookie scoop or spoon, portion the dough and roll into 1-inch balls. Place the balls 1 inch apart on the prepared baking sheet. Bake for 15-20 minutes, until the cookies are just set but not yet brown. Transfer to a wire rack and let cool completely (if you would like to roll some of the cookies in powdered sugar to make a traditional version of snowball cookies without the chocolate and peppermint, do so now while the cookies are still warm. Leave any cookies you'd like to top with chocolate plain and let cool.)
With an electric mixer, beat the butter, powdered sugar, vanilla extract, and peppermint extract in a large mixing bowl until light and fluffy. Beat in the flour and salt, then the nuts until evenly blended. Divide the dough in half, form each half into a disk, then wrap tightly in plastic and refrigerate for 30 minutes.