In a small bowl, whisk the flour with the salt. In a medium bowl, using an electric mixer, beat the butter with the sugar at medium speed until creamy, about 3 minutes. Beat in the egg and vanilla. Add the flour and beat at medium-low speed until a very soft dough forms. Divide the dough in half, flatten into disks and wrap in plastic. Refrigerate the dough until firm, about 30 minutes.
Line 2 large baking sheets with parchment paper. On a well-floured surface, roll out one disk of dough 3/8 inch thick. Using a 2 1/4-inch round cutter, stamp out as many cookies as possible; transfer to the baking sheets. Repeat with the second disk of dough. Gather, chill and reroll the scraps, cutting out more cookies. Refrigerate for 15 minutes, until firm.
Preheat the oven to 350°. Bake the cookies for about 18 minutes, until golden, shifting the pans halfway through baking. Let the cookies cool for 5 minutes, then transfer them to racks to cool completely.