Add the flour, baking powder, nutmeg and salt to a fine-mesh sieve and sift into a large bowl. In the bowl of a stand mixer fitted with the paddle attachment add the sugar and shortening and cream on medium-low speed until combined. Increase the speed to medium and beat until the mixture is light and fluffy, 1 to 2 minutes. Add the eggs, one at a time, incorporating each one completely before adding the next addition. Add the milk and vanilla, then reduce the mixer speed to low and add the dry ingredients and mix just until a smooth dough comes together.
Generously coat the work surface with flour and turn the dough out onto the floured surface. Dust the top of the dough with more flour, then roll the dough into a 1/2-inch thick sheet. Use a 2 1/2 to 3-inch doughnut cutter to stamp out circles from the dough. Place the circles on a parchment paper-lined baking sheet. Bring the scraps together, re-roll them into a 1/2-inch thick sheet, and stamp as many circles from the dough as possible (discard the remaining scraps); you should get about 2 dozen doughnuts.
Line a rimmed baking sheet with paper towels and set a wire cooling rack on top. Pour the oil into a large saucepan set over medium-high heat; the oil should fill the saucepan by about 3 inches-if it doesn't, add more. Heat the oil until it reads 375 degrees F on an instant-read thermometer. Add 2 to 3 doughnuts to the hot oil and cook until the doughnuts are golden brown, 1 1/2 to 2 1/2 minutes on each side. Transfer to the cooling rack to drain and cool and repeat with the remaining doughnuts, making sure the temperature of the oil returns to 375 degrees F between batches.
Serve the doughnuts plain or roll in cinnamon-sugar or confectioners' sugar before serving.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.