In a large bowl, sift together flour, sugar, salt, baking powder, cinnamon and nutmeg. Mix in butter until crumbly.
Stir in milk and egg until smooth. Knead lightly, then turn out onto a lightly floured surface. Roll or pat to 1/2cm thickness. Cut with a doughnut cutter, or use two round biscuit cutters of different sizes.
Carefully drop doughnuts into hot oil, a few at a time. Do not overcrowd pan or oil may overflow. Fry, turning once, for 3 minutes or until golden. Drain on paper towels.