In a large bowl, stir together the flour, 1/2 cup sugar, baking powder, salt, 1 teaspoon of cinnamon and nutmeg. Make a well in the center and pour in the milk, egg, butter, and vanilla. Mix until well blended. Cover and refrigerate for 1 hour.
Heat oil in a deep heavy skillet or deep-fryer to 370 degrees F (185 degrees C). On a floured board, roll chilled dough out to 1/2 inch thickness. Use a 3 inch round cutter to cut out doughnuts. Use a smaller cutter to cut holes from center. If you do not have a small cutter, use the mouth of a bottle.
Fry doughnuts in hot oil until golden brown, turning once. Remove from oil to drain on paper plates. Combine the remaining 1/2 teaspoon cinnamon and 1/2 cup sugar in a large resealable bag. Place a few warm donuts into the bag at a time, seal and shake to coat.