To make the buttermilk (or soured milk), combine the milk and vinegar and set aside for 10 minutes; it will curdle and thicken.
Sift flour, baking powder, baking soda, salt and allspice together.
In a large mixing bowl, cream the shortening and sugar together. Beat in the egg and vanilla.
Stir in the flour mixture and buttermilk alternately in three batches, starting with the flour, and mix to a soft dough. Cover the bowl and refrigerate for at least 1 hour but no more than 6 hours.
On a lightly floured surface, roll out dough to 1cm thick. Use a doughnut cutter to cut out 6cm rounds and the holes. Alternatively, use cookie cutters in two sizes.
Pour about 6cm of oil into a medium saucepan and heat until simmering (a deep-fry thermometer will show 190°C). Fry the doughnuts and holes until golden, 20-30 seconds on each side. Remove from oil onto a plate lined with kitchen paper. Dust with snow or icing sugar and serve.