In a bowl, combine flour and baking powder. In a separate bowl, mix together butter and sugar then stir until fluffy. Add in eggs and vanilla. Slowly stir in the flour mixture. Chill in the fridge for an hour.
Preheat oven to 180 degrees C. Line a baking tray with baking paper. Form into tablespoon sized balls, place on baking tray then press down with a fork or the flat bottom of a glass. Bake biscuits in preheated oven for 10 minutes, until golden. Allow to cool on a cooling rack before icing.
Mix togther sugar, butter and vanilla. Slowly stir in the milk until smooth, stopping when a spreadable texture is reached.
If you would like to dye the icing, you can add food colouring until the desired colour is reached.
For fancy shaped sugar cookies, roll biscuit dough on a lightly floured surface then cut out shapes with cookie cutters.