To make the dough, mix together the butter and sugar using a hand mixer, then add the egg and mix to get a smooth paste.
Add the flour and vanilla extract and knead well to get an elastic dough.
Cover the dough with clingfilm and refrigerate for 30 minutes.
Roll the dough on the work surface, and use a star-shaped cookie cutter to shape 30 stars.
Arrange onto 2 baking trays lined with baking paper, cover the trays with clingfilm, then chill in the fridge for one hour.
Preheat the oven to 180 degrees Celsius (350 Fahrenheit).
Bake the cookies one tray at the time for 8-10 minutes, they will still have a pale yellow colour, but will be firm to touch.
Arrange on a plate and leave to cool completely.
To make the icing, sift the sugar, add the milk and vanilla extract and mix well to get a smooth paste.
Fill a piping bag with the icing sugar, and using a small round nozzle, ice every cookie, then sprinkle with any festive sprinkles or decorations.