1/2 cup confectioners' sugar, plus more for dusting
1 1/2 tablespoons pure vanilla extract
2 1/4 cups all-purpose flour
Pinch of salt
Preheat the oven to 350° and position one rack in the upper third and one in the lower third. Line 2 large baking sheets with parchment paper.
In the bowl of a standing electric mixer fitted with the paddle, beat the butter with 1/2 cup of the confectioners' sugar until pale white, about 5 minutes. Beat in the vanilla. Add the flour and salt and beat at low speed just until combined.
On a lightly floured surface, roll level tablespoons of the dough into 3-inch ropes. Taper the ends slightly and form the ropes into crescents. Carefully transfer the crescents to the baking sheets, about 1/2 inch apart.
Bake the crescents for 22 to 24 minutes, until the bottoms are golden and the tops are pale blond; shift the baking sheets from top to bottom and front to back halfway through for even baking. Transfer the baking sheets to racks and let the cookies cool for 10 minutes.
Fill a small bowl with confectioners' sugar. While the cookies are still warm, coat them in the sugar and transfer to a clean sheet of parchment paper to cool slightly. Roll the cooled cookies in the sugar again and let cool completely.
The crescent cookies can be stored in an airtight container between sheets of wax paper for up to 1 week. Dust the cookies very lightly with confectioners' sugar before serving.