Beat butter at medium speed with an electric mixer until creamy; gradually add sugar, beating well. Stir in vanilla.
Combine flour and salt; gradually add to butter mixture, beating at low speed until blended.
Roll dough to 1/2" thickness on a lightly floured surface. Cut with a 2 1/2" round cutter or Christmas cookie cutter. Place 2" apart on an ungreased baking sheet.
Bake at 275° for 50 minutes. Cool 2 minutes on baking sheet. Remove to a wire rack to cool.
Note: Unbleached flour is the best-quality, freshest flour for baking and is particularly good in simple recipes like shortbread. Other kinds of all-purpose flour work fine in this recipe, too.