Sift together flour, cake flour, baking powder, and baking soda. Whisk in nutmeg and salt. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, mix together egg and sugar. Add buttermilk and shortening. Mix on medium speed until well combined.
Add dry flour mixture. Mix on low speed until there are no longer any dry streaks of flour. Batter will be thick and a little sticky. Transfer batter to a small bowl. Cover with plastic wrap and let chill in the fridge for 1 hour (up to overnight).
Fill a large pot with 3 inches of vegetable oil. Using a candy thermometer, bring oil to 360°F over medium heat.
Remove chilled dough from the fridge. Place on a floured work surface. Dough will be sticky! Lightly sprinkle top of dough with flour. Roll or pat dough to a thickness of about ½-inch. Flour a 3-inch round cutter. Punch out rounds. Fold and press together dough scrapes to punch out additional rounds. Use a 1-inch or smaller round cutter to punch out doughnut holes.*Flour cutters as needed to prevent dough from sticking. Brush off any excess flour from donuts before adding them to the hot oil. Depending on the size of your pot, add 3-5 donuts to the pot at a time. Do not overcrowd the donuts, as it will decrease the oil temperature too much.
Set a timer for 75 seconds. By this time, the donuts should have floated to the top. Turn donuts over and cook other side for another 75 seconds. Once golden brown all over, remove donuts from oil. Let drain on paper towels or over wire rack.
Repeat for doughnut holes, reducing time to 50-60 seconds per side.
Allow donuts to cool for 5 minutes. Coat in cinnamon sugar mixture until donuts are still warm.