In the bowl of an electric mixer fitted with the paddle attachment, combine butter and light-brown sugar; beat on high speed until fluffy, 2 to 3 minutes. Add egg and vanilla, and mix to combine. Add flour and salt, and mix on low speed until flour is incorporated.
Roll dough into three 1 1/2-inch-diameter logs. Wrap in plastic wrap, and refrigerate until firm, at least 1 hour or overnight.
Heat oven to 350 degrees. Line baking sheet with parchment paper or a Silpat. Roll cookie log in sugar, coating it evenly, and slice into 1/4-inch rounds. Place cookies on baking sheet about 1 inch apart. Using a cake tester or toothpick, make four decorative holes in each cookie. Bake until golden brown, 15 to 20 minutes. Remove from oven, and let cool completely on wire racks.