Preheat the oven to 375 degrees, and line baking sheets with parchment.
Using the paddle attachment, mix the butter and sugar, just until combined. Do not cream this mixture.
Add the egg and mix again
Add the flour, cornstarch, vanilla and salt, and mix on medium low speed. The mixture will seem very dry and sandy at first, but after 3 to 5 minutes in the mixer it should gather itself into a ball and pull away cleanly from the sides of the bowl. (See Note 1)
Roll the dough out between 2 sheets of parchment paper, to a thickness of 1/4-inch to 1/2-inch. (See Note 2)
Cut into shapes, peel away from the bottom sheet of parchment carefully and flip onto the prepared baking sheets. (See Note 3)
Bake until set around the edges but still a bit soft in the centers. Check on the 7-8 minute mark. A slight browning on the sides mean it's done (See Note 4)