Heat oil in a deep fryer to 375°F (190°C).

Sift together flour, sugar, baking powder, salt, cinnamon, and nutmeg in a large bowl.

Mix in melted butter until crumbly.

Stir in milk and beaten egg until smooth.

Knead lightly, then roll or pat dough to 1/4-inch thickness on a floured surface.

Cut dough using a doughnut cutter or two different-sized biscuit cutters.

Lay doughnuts in hot oil, a few at a time, without overcrowding. Fry, turning once, until golden brown (about 3 minutes).

Drain fried doughnuts on paper towels.