Whisk together cake flour, all-purpose flour, baking powder, salt, nutmeg, and cinnamon in a medium bowl. Set aside.
In the bowl of stand mixer fitted with a paddle attachment, mix together sugar and shortening on low speed. Mix until mixture resembles coarse sand. Add whole egg and yolk. Mix until creamy. Scrape down bowl as needed to ensure thorough mixing.
Add 1/3 of dry mixture into mixing bowl. Mix on low speed until almost combined. Following with 1/2 of milk. Add another 1/3 of dry mixture, followed with remaining milk. Mix and end with dry mixture.
The dough will be quite sticky. Once combined, transfer dough to a clean bowl. Cover directly with plastic wrap. Refrigerate for one hour.
Filled a large pot with 3 inches of oil. Using a candy thermometer, bring oil to 370 degrees F over medium heat.
Remove dough from fridge and transfer to generously floured surface. Dough will be very sticky. Sprinkle cake flour over dough as needed to ensure dough doesn't stick to surface. Pat dough into an even layer about 1/2-inch thick.
Dip 2 1/2-inch round cutter in flour each time before cutting dough. Cut smaller doughnut hole in center. Fold and press dough scrapes together to cut out more doughnuts.
Brush off any excess flour off the doughnuts before adding them to the oil hot. Depending on the size of your pot, add 4-5 doughnuts to the pot at a time. Be sure not to crowd the doughnuts. Set a timer for 1 minute and 30 seconds. By this time, the doughnuts should have floated to the top. Turn over doughnut and cook for another 90 seconds. Once golden brown all over, remove from oil and let drain on paper towels or over wire rack.
Allow doughnuts to cool before dipping in pink icing.