Prepare the Donut Pan: Preheat the oven to 350°F. Lightly grease a donut pan with nonstick cooking spray or butter, making sure the cavities are well coated.
Prepare the Apple: Peel and finely dice the apple into small pieces. Smaller pieces distribute more evenly throughout the donut batter.
Combine the Dry Ingredients: In a mixing bowl, whisk together the flour, baking powder, cinnamon, nutmeg, and salt until evenly combined.
Mix the Wet Ingredients: In a separate bowl, whisk together the melted butter, granulated sugar, brown sugar, eggs, vanilla extract, and apple cider until smooth.
Make the Donut Batter: Gradually add the dry ingredients to the wet ingredients. Stir gently until just combined, then fold in the diced apple. Do not overmix the batter.
Fill the Donut Pan: Transfer the batter to a piping bag or zip-top bag with the corner snipped off. Pipe the batter evenly into the prepared donut cavities, filling each about three-quarters full.
Bake the Donuts: Bake for 12–15 minutes, or until the donuts are lightly golden and spring back when gently touched. A toothpick inserted into a donut should come out clean or with a few moist crumbs.
Cool the Donuts: Let the donuts cool in the pan for a few minutes, then carefully transfer them to a wire rack. Allow them to cool before glazing.
Make the Apple Cider Glaze: Whisk the powdered sugar, apple cider, and cinnamon together in a small bowl until smooth. Add a little more powdered sugar or cider if needed to reach a thick but pourable consistency.
Glaze the Donuts: Dip the tops of the cooled donuts into the apple cider glaze. Let the excess glaze drip off, then place the donuts back on the wire rack.
Serve and Enjoy: Allow the glaze to set slightly before serving. Enjoy the donuts fresh with coffee, tea, warm apple cider, or your favorite breakfast drink.