Put a rack in the middle of the oven and preheat to 375 degrees F.
Blend together butter, sugar, vanilla and salt in a food processor (or bowl) until well combined. Sift flour over butter mixture and blend until a soft dough forms.
Transfer dough to an ungreased baking sheet. With floured fingertips, pat into a 9-by-4 1/2 inch rectangle. Crimp edges decoratively with tines of fork and prick dough all over. Score dough crosswise with the back of a knife into 8 pieces.
Bake shortbread until edges are golden, about 15 minutes.
Cool on baking sheet on a rack for 10 minutes, then cut into slices (while still warm) with a sharp knife. Transfer to a rack to cool completely.
Keeps in an airtight container at room temperature for up to 1 week.