Beat confectioners' sugar and margarine together in a bowl using an electric mixer until smooth and creamy. Add vanilla extract and beat until smooth.
Whisk 2 1/2 cups flour, cornstarch, baking powder, and baking soda together in a separate bowl. Beat flour mixture into creamed butter mixture on medium-low speed just until dough is combined. Separate dough into 2 balls and refrigerate for at least 2 hours.
Preheat oven to 350 degrees F (175 degrees C).
Dust dough with flour and roll into 1/4-inch thick pieces on a work surface. Cut cookies into desired shapes using cookie cutters and arrange on a baking sheet.
Bake in the preheated oven until cookies are just beginning to turn brown, 10 to 14 minutes. Let cookies rest on baking sheet, 5 to 10 minutes; transfer to a wire rack to cool.
