Preheat the oven to 180°C/fan 160°C/gas mark 4. Line 2 baking trays with baking parchment. Beat the butter and sugar together in a bowl until pale and fluffy. Add the egg and continue to beat.
Sift in the flour, then add the vanilla extract and fold through to form a dough. Turn out the dough on to a lightly floured surface and knead gently until smooth. Chill in the fridge for 30 minutes.
Remove the dough from the fridge and roll out until it's about 4mm thick. Use a 7cm bunny-shaped biscuit cutter to cut biscuits from the dough. Arrange on the baking trays and bake for 12-15 minutes, until golden. Remove from the oven and leave the biscuits on the trays to cool slightly before transferring to wire racks to cool completely.
Meanwhile, combine the icing sugar, milk and food colouring in a bowl until smooth. Carefully spread over the biscuits and press a marshmallow tail on each.