Doughnuts: Boil 1 cup apple cider in a small pot until reduced to 1/4 cup, about 10 minutes. Remove pot from heat and cool to room temperature.
Beat white sugar and butter together with an electric mixer until creamy and smooth then add eggs, 1 at a time, beating well before each addition. Mix buttermilk and reduced apple cider into creamed butter mixture.
Whisk flour, baking powder, baking soda, cinnamon, salt, and nutmeg together in a bowl. Slowly stir flour mixture into creamed butter mixture just until dough is combined.
Turn dough onto a lightly floured piece of baking paper. Sprinkle dough with flour and roll to 1cm thickness, adding more flour if dough is sticky. Cut dough into doughnuts using a 10cm wide doughnut cutter and arrange on a baking tray. Refrigerate until chilled, 20 to 30 minutes.
Heat at least 10cm of oil in a deep-fryer or large saucepan to 180 degrees C.
Glaze: Whisk icing sugar and 2 tablespoons apple cider together in a bowl until glaze is smooth.
To Serve: Fry doughnuts in the hot oil, 2 to 3 at a time, until golden brown, 1 to 2 minutes per side. Remove with a slotted spoon and drain on paper towels. Pour glaze over warm doughnuts.