Put the flour and cornflour in a mixing bowl or a food processor. Add the butter and rub in with your fingertips or process until the mixture resembles fine breadcrumbs. Stir in the sugar and vanilla until mixed, then squeeze the crumbs together with your hands to form a smooth ball.
Knead lightly then roll out thinly on a lightly floured surface. Stamp out festive shapes using biscuit cutters and transfer to ungreased baking sheets. Reknead the trimmings and continue rolling and stamping out until all the dough has been used.
Prick the shapes with a fork, then bake in a preheated oven, 180°C (350°F), Gas Mark 4, for 10 minutes or until pale golden. Leave to cool on the baking sheet.
Make the icing. Place the egg white in a bowl. Gradually mix in the icing sugar and lemon juice to give a smooth consistency. Add extra water if the icing seems too thick. Divide between 2 or more bowls and colour as you like.
Spoon the icing into greaseproof paper piping bags, snip off the tips and pipe outlines around the edge of the cookies. Leave to harden for 10 minutes. Fill in the rest of the surface of the cookie with the same colour icing to create a smooth evenly covered cookie top. Leave to dry. Finally, pipe white icing over the top of the coloured surface to outline and make specific features.