In a small saucepan, simmer the apple cider over medium heat until reduced from 1 ½ cups to ½ cup--this will take about 15-20 minutes. Set aside to cool.
Preheat the oven to 350°F (175°C) and grease your donut pans.
In a large bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt.
In a separate bowl, whisk together the melted butter, brown sugar, vanilla extract, eggs, milk, and cooled reduced apple cider until smooth.
Pour the wet ingredients into the large bowl of dry ingredients and gently fold until just combined.
Stir in the sourdough discard until the batter is smooth—do not overmix.
Transfer the batter to a piping bag and pipe evenly into the prepared donut pans.
Bake for 10–13 minutes, or until the donuts spring back when lightly pressed.
Let the donuts cool in the pan for 5 minutes before transferring to a wire rack.
For the coating, brush each warm donut with melted butter, then roll in the cinnamon sugar mixture until fully coated.