In a bowl, sift together the flour, salt, baking soda, and cream of tartar. Set aside. In the bowl of a heavy-duty mixer fitted with the paddle, cream the butter and shortening on medium speed until light and fluffy. Add the confectioners’ sugar and beat on low speed until smooth. Add the egg and vanilla extract and continue beating on low until well combined.
On low speed, slowly add the reserved dry ingredients. Beat until well combined. Cover with plastic wrap. Transfer to the refrigerator and chill until very firm, at least 2 hours.
Preheat the oven to 350°F. Using an ice cream scoop or a melon baller, take some of the dough and shape it into small 1-inch balls. (Keep the remaining dough in the refrigerator, as it must be kept chilled.) Place the dough balls on an ungreased baking sheet and press with the tines of a fork to flatten. Bake until pale golden, but not brown, 10 to 12 minutes. Cool slightly on the baking sheet on a rack. Then, using a spatula, transfer the cookies to a rack to cool completely.