Preheat your oven to 350°F (175°C) and line two large baking sheets with parchment paper.
In a large bowl, use an electric mixer on medium speed to beat the softened butter and 1/2 cup of powdered sugar together until the mixture is smooth, light, and creamy, scraping down the sides of the bowl as needed. Beat in the vanilla extract.
In a separate medium bowl, whisk together the flour and salt. Gradually add the flour mixture to the butter mixture on low speed, mixing just until the dough starts to come together. Use a spatula to fold in the finely chopped pecans until they are evenly distributed and no dry streaks of flour remain. The dough will be slightly crumbly but should hold together when pressed.
Scoop the dough by the level tablespoon and roll it between your palms into 1-inch balls. Place the balls about 2 inches apart on the prepared baking sheets; they will not spread much.
Bake for 12 to 15 minutes, or until the bottoms are just lightly golden and the tops are still pale and look set. The cookies will be very soft.
Let the cookies cool on the baking sheet for 5 minutes, until they are firm enough to handle but still warm. Carefully roll each warm cookie in the remaining 1 cup of powdered sugar to coat it lightly. Place the cookies on a wire rack to cool completely. Once cooled, roll them in the powdered sugar a second time for a thick, snowy coating.
