Finely chop the pecans. In the bowl of a stand mixer fitted with a paddle attachment, beat the butter and powdered sugar together on low speed to blend. Increase speed to medium and continue to beat until soft and light. Add the chopped pecans and beat until distributed, scraping down the sides of the bowl as needed. Add the salt, vanilla, and flour and beat on low speed to blend. Scrape down the inside of the bowl and beat again on low speed until dough is thoroughly combined.

Line two baking sheets with parchment paper.

Scoop the dough into 30 portions, using about 1 tablespoon for each, and shape each into a ball. Arrange on the prepared baking sheets 1 inch apart. Transfer the baking sheets to the refrigerator to chill while the oven preheats.

Preheat the oven to 300°F.

Bake the cookies on upper-third and lower-third racks of oven for 15 minutes, then switch pan positions. Continue baking until cookies feel firm to the touch and are golden on the undersides, about 15 minutes.

Check to see that cookies are done. Remove from oven or add time as needed.

Let cookies cool on the baking sheets for 5 minutes. Place the powdered sugar in a medium bowl. Gently toss about four warm cookies at a time in the powdered sugar to thoroughly coat. Transfer to a cooling rack and let cool completely, about 30 minutes.

Roll the cooled cookies in powdered sugar a second time. Serve immediately. The cookies can be stored in an airtight container for up to 1 week, or frozen for up to 1 month.