In a bowl of an electric mixer, cream the butter and brown sugar on high speed until light and fluffy, about 2 to 3 minutes.
Add the egg and vanilla, mixing thoroughly.
Gradually add the flour and salt. Mix on low speed until just combined.
On three separate sheets of plastic wrap, roll the dough into three logs 1 1/2-inches thick. Wrap in plastic and refrigerate for at least one hour to overnight.
When ready to bake, place a rack into the center of the oven and preheat to 350 degrees F.
Prepare a cookie sheet by lining it with parchment.
Remove one log from the fridge and roll it in the granulated sugar, making sure to coat it completely.
With a sharp knife, cut slices 1/4-inch thick.
Place the slices on the prepared cookie sheet 1-inch apart.
Bake for about 15-20 minutes, or until lightly golden brown.
Transfer cookies to a wire rack so that they can cool completely.