In a large, deep heavy pot or an electric fryer heat the oil to 375 degrees.
In a large bowl, beat together the egg and sugar. Stir in the milk and the shortening. Sift together all of the dry ingredients. Stir the dry ingredients into the egg mixture and stir to combine. Chill the dough for 30 minutes. On a lightly floured surface, flatten the dough to 1/2 inch thickness with your fingertips or a rolling pin. Cut out the doughnuts using a doughnut cutter or two sizes of round cookie cutters. Transfer doughnuts to wax paper and allow to dry for 10 minutes. Using a spatula dipped in the hot oil, carefully transfer the doughnuts to the hot oil. Cook 2 - 3 doughnuts at a time, flipping them when they brown on a side. Remove to paper towels to drain. Dust with confectioners' sugar or cinnamon sugar mixture. Serve warm.
Doughnut holes: Add dough to hot oil by the spoonful and fry until golden.
Variations: Chocolate doughnuts: Combine the melted shortening with 1 ounce of melted chocolate before adding to batter.
Glazed doughnuts: Combine 1/2 cup confectioner's sugar with 2 teaspoons hot milk and 1/2 teaspoon vanilla. Drizzle over cooled doughnuts.
Jelly doughnuts: Stir 1/2 cup jelly of choice until smooth. Transfer jelly to a plastic baggie and snip off one corner to squeeze out jelly. Poke doughnut round with a chopstick And squirt in jelly.