Grease and line 2 baking sheets with nonstick baking paper. Beat together the butter and sugar until pale and fluffy. Gradually beat in the egg and vanilla extract adding a little flour to prevent the mixture curdling. Add the flour and fold in to make a stiff dough. Wrap in clingfilm and refrigerate for 20 minutes.
Roll out the dough on a lightly floured surface to 2.5 mm (⅛ inch) thick. Cut out about 30 rounds, using a 4–5 cm (1½–2 inch) diameter cookie cutter, rerolling the trimmings as necessary. Place on the baking sheets.
Bake in a preheated oven, 180°C (350°F), Gas Mark 4, for 5 minutes, or until a pale golden colour. Transfer to a wire rack and leave to cool. Roll out the white icing on a sugar-dusted surface until 2.5 mm (⅛ inch) thick. Using the same cutter as before, dusted this time with a little icing sugar, cut out the same number of circles. Using a clean paintbrush, paint a little of the sieved jam on each cookie to ensure that the icing will stick, place the icing shapes on the cookies and press down lightly. Using the coloured writing icing, write a letter on each cookie to spell out the name or message.