In a bowl with an electric mixer, beat the butter and sugar until fluffy. add the egg and beat until light and fluffy, fully incorporated. Add the vanilla and beat.
In a separate bowl, sift the flour and salt together. Gradually add to the egg mixture, until just incorporated.
Line a sheet pan with parchment. Place the dough on the pan and cover with another sheet of parchment. Roll the dough between the two sheets until approximately 1/4 inch thick. Place in the fridge to chill for at least 1/2 hour.
Preheat the oven to 350 degrees. using cookie cutters of your choice, cut the cookies and place on a second cooking sheet lined with parchment, about 2" apart.
Bake for about 10 minutes until just beginning to brown around the edges. Cool completely and decorate if desired. If you are not going to ice, you can just sprinkle with sugar before baking.
Royal icing makes these cookies fun to decorate. A good icing recipe, 2 egg whites, or the equivalent of powdered egg whites, and 1/4 teaspoon of cream of tartar. Beat the egg whites until just frothy, add 3 cups of powdered sugar, 1 teaspoon of lemon juice. Using a beater or the whisk attachment on your mixer, beat the icing at high speed until it is very creamy and white. This may take a couple minutes. Color and use.
To ice cookies, use a plastic bag, place the icing in the bag and cut a corner just a tiny bit. Outline the cookies with the color. Allow to dry. Then thin the icing and "flood" inside the outline.