In the bowl of a stand mixer fitted with a paddle attachment or a large bowl, beat together the butter and sugar until light and fluffy.
Mix in the orange zest, orange juice, and vanilla extract.
In a medium bowl, combine the flour, cornstarch, and salt. Mix into the butter just until combined.
Form the dough into a disc and wrap in plastic. Refrigerate until well chilled and firm, about 1 hour.
Preheat oven to 350˚F (180˚C). Line two baking sheets with parchment.
On a lightly floured surface, roll the dough into a sheet 1/4 inch (0.5 cm) thick. Cut the dough into shapes using desired cookie cutters. Transfer to prepared baking sheets 1 inch (2.5 cm) apart.
Bake in preheated oven until the bottoms are lightly golden, 12-15 minutes depending on the size of the cookies. If the cooking is uneven, switch and rotate the baking sheets halfway through the time.
Allow to cool on the baking sheets for 5 minutes before transferring to a wire rack to cool to room temperature.