In a medium bowl, whisk together the flour, cornstarch, baking soda, cream of tartar and salt, and set aside. Using a stand or hand mixer, cream together the butter and sugar. Next beat in the egg and the vanilla, followed by slowly mixing in the dry ingredients.
Wrap the dough in wax paper and chill for at least an hour.
Roll out the dough one-eighth to one-half inch thick. Cut with your favorite cookie cutter and bake for six to eight minutes (or longer depending on size of cookies.) Remove cookies from the oven when they are just barely light brown around the edges. Allow to cool for several minutes on the cookie sheet before transferring to cooling rack.