Pour vegetable oil into a large pot to a depth of about 2 inches and place over medium heat until oil reaches 360◦. Place two cooling racks over two large rimmed baking sheets lined with paper towels.
Mix 1 cup of the sugar and 2 tablespoons cinnamon in a shallow bowl and set aside.
In a large bowl, whisk the flour, baking powder, salt, remaining 1/4 teaspoon cinnamon, and nutmeg.
In a medium bowl, whisk the eggs and remaining 1 1/2 cups sugar until smooth. Whisk in the milk and shortening.
Add the egg mixture to the bowl with the flour mixture and use a large wooden spoon to stir until fully incorporated. (The dough will be a bit sticky.)
Turn the dough out onto a well floured surface and roll to 1/2-inch thick, sprinkling the top with a little more flour if the rolling pin sticks. Use a donut cutter to cut out donuts, dipping the cutter into flour between each cut. Re-roll the scraps once so you end up with 2 dozen donuts. Place the donuts on a large platter or cutting board, setting aside the holes.
Working in batches of 4 to 6, depending on your pan size, carefully lower the donuts into the hot oil using a spider. Cook until browned on the bottom, about 2 minutes, then gently turn over and cook until the other side is browned, another 1 to 2 minutes. Remove the donuts from the oil with the spider and place them on the cooling rack. As soon as you can safely handle the hot donuts, dip them in the cinnamon and sugar mixture, fully coating both sides, then return them to the cooling rack. Repeat with the remaining batches, waiting for the oil to heat up to 360◦ again if needed. Once the donuts are done, you can also cook the holes if you'd like.
As soon as the donuts are cool, enjoy - they are at their best fresh!