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 1/4 inch thickness. Cut with a doughnut cutter, or use two round biscuit cutters of different sizes.
Carefully place doughnuts into hot oil, a few at a time. Do not overcrowd pan or oil may overflow. Fry, turning once, for 3 to 5 minutes or until golden. Drain on inverted cooling rack over paper towel-lined sheet tray to wick excess oil away.