Heat oil in a deep-fryer or large saucepan to 190 degrees C.
Doughnut: Whisk buttermilk, white sugar and eggs together in a bowl. Mix flour, baking soda, baking powder, salt, nutmeg and cinnamon together in another bowl. Stir buttermilk mixture into flour mixture until combined then add butter and knead until a soft dough forms.
Turn out dough onto a lightly floured work surface and roll to 6mm thick. Cut dough into doughnut shapes using a 6cm doughnut cutter.
Glaze: Beat icing sugar, margarine and vanilla extract together in a bowl until smooth. Gradually add milk, stirring constantly, until desired glaze-consistency is reached.
To Prepare: Working in batches, cook doughnuts in hot oil until golden brown, about 1 minute per side. Transfer cooked doughnuts to a paper towel lined plate to drain. Dip hot doughnuts into glaze to coat.