Preheat oven to 350 degrees F.
Beat butter, sugar, and salt with an electric mixer at moderately high speed until pale and fluffy, about 2 minutes. Add egg and vanilla and mix at low speed until just combined. Add flour and mix until just combined.
Drop level teaspoons of batter about 3 inches apart on buttered baking sheets. Bake cookies in batches in middle of oven until just golden around edges, 6 to 8 minutes. Cool on baking sheet on a rack 1 minute then transfer with a metal spatula to rack to cool completely.
Note: Cookies can be made 3 days ahead and kept, layered between sheets of waxed paper in an airtight container, at room temperature.