Melt the shortening in a small saucepan over medium heat. Set aside to cool slightly but still liquid.
In a standing mixer fitted with the paddle attachment, beat the sugar and shortening together on medium speed, until just combined. Add the eggs, one at a time, beating well after each addition. Add the milk and vanilla, and continue beating until the mixture is light, about 2 minutes.
Meanwhile, sift the flour, baking powder, nutmeg, mace, and salt into a medium bowl.
Reduce the mixer's speed to low, add the flour mixture until just combined. (The texture of the dough will be soft and loose.) Transfer the dough to a large piece of plastic wrap and wrap it well. (Make sure you scrap off all the dough clinging to the paddle.) Refrigerate the dough for 4 hours or overnight.
On a lightly floured work surface, roll the chilled dough out into a rectangle about 1/2-inch thick. Dip the edges of a 2 1/2-inch-round doughnut cutter in flour and cut the dough into doughnuts. Place the doughnuts and holes on a baking sheet lined with parchment.
Put enough vegetable shortening into a tall, heavy-bottomed pot to fill it about a third of the way up. Heat the shortening over medium heat until a deep-fry thermometer inserted in the oil registers 375 degrees F. Line a baking sheet with paper towels.
Working in batches, fry the doughnuts and holes, turning once, until they are golden brown, about 2 to 3 minutes. Using a slotted spoon, transfer the doughnuts to the paper towels to drain and cool. Repeat until all the doughnuts and holes are fried. (Make sure the shortening returns to 375 degrees F. between batches.)
For sugared doughnuts, roll the doughnuts in sugar or cinnamon sugar while they are still warm. To cover the doughnuts with confectioners' sugar, allow them to cool completely before rolling in the sugar.
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