In a medium bowl, using an electric mixer, beat the butter with the 1/3 cup of sugar and the salt at medium speed until light and fluffy, about 3 minutes. Beat in the vanilla, then beat in the flour at low speed, scraping the side and bottom of the bowl, until the dough just comes together. Add the pecans and beat just until they are incorporated and lightly broken up. Divide the dough in half and form it into two 2-inch-thick logs. Wrap tightly in plastic and refrigerate overnight.
Preheat the oven to 350°. Line 3 baking sheets with parchment paper. Working with 1 log at a time and keeping the other one chilled, cut the dough into scant 1/4-inch-thick slices, arrange them on the baking sheets and sprinkle them with sugar. Repeat with the second log of dough.
Bake for 25 to 30 minutes, until the cookies are lightly golden around the edges and on the bottom, shifting the baking sheets halfway through. Let the cookies cool on the baking sheets for a few minutes, then transfer them to a rack to cool completely.
The cookies can be stored in an airtight container for up to 3 days.