Preheat your oven to 350°F. Line a cookie sheet with parchment paper.
In a medium bowl, beat the softened butter with the vanilla extract and milk until smooth.
Add the all-purpose flour, salt, and powdered sugar. Stir until the dough just comes together. If the dough is too crumbly, add 1 teaspoon of milk at a time until it comes together.
Stir in the finely chopped walnuts until evenly combined.
Take small scoops of dough and roll them between your hands into 1-inch balls. Place them about 2 inches apart on the prepared cookie sheet.
Bake in the preheated oven for about 11-13 minutes, or until the edges just begin to turn light golden.
Transfer the cookies to a cooling rack and let them cool for about 15 minutes. While still slightly warm, roll each cookie in additional powdered sugar until well coated.
Allow the cookies to cool completely on the rack. Once cooled, store them in an airtight container.