In a large bowl, whisk together the all-purpose flour, granulated sugar, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg.

In a separate medium bowl, whisk together the apple cider, large egg, melted unsalted butter, and vanilla extract.

Gradually add the wet ingredients to the dry ingredients, mixing until just combined. Be careful not to overmix the dough.

Cover the dough with plastic wrap and refrigerate for at least 30 minutes. This will make the dough easier to handle.

On a lightly floured surface, roll out the chilled dough to approximately 1/2-inch thickness. Use a donut cutter to cut out the donuts and the smaller donut holes.

In a large, heavy-bottomed pot or deep fryer, heat the vegetable oil to 350°F (175°C). Use a thermometer to ensure accurate temperature.

Carefully place a few donuts at a time into the hot oil, ensuring not to overcrowd the pot. Fry for 1-2 minutes per side, until they are golden brown and cooked through.

Using a slotted spoon, remove the fried donuts from the oil and place them on a wire rack lined with paper towels to drain any excess oil.

In a shallow dish or plate, combine the powdered sugar and an additional teaspoon of ground cinnamon for the coating.

While the donuts are still warm, gently toss them in the cinnamon sugar mixture until they are evenly coated on all sides.

Serve the apple cider donuts warm and enjoy your homemade treat!