Beat icing sugar and margarine together in a bowl using an electric mixer until smooth and creamy. Add vanilla extract and beat until smooth.
Whisk 300g plain flour, cornflour, baking powder and bicarbonate of soda together in a separate bowl. Beat flour mixture into creamed butter mixture on medium-low speed just until mixture is combined. Separate cookie dough into 2 balls and refrigerate for at least 2 hours.
Preheat oven to 180 C / Gas 4.
Dust dough with flour and roll cookie dough out to 1/2cm thick on a work surface. Cut cookies into desired shapes using cutters and arrange on a baking tray.
Bake in the preheated oven until sugar cookies are just beginning to turn brown, 10 to 14 minutes. Let cookies rest on baking tray, 5 to 10 minutes; transfer to a wire rack to cool.