Bring the apple cider to a gentle boil in a small saucepan. Simmer over medium heat until reduced to ⅓ cup, 20 to 30 minutes. Set aside to cool slightly.
Preheat the oven to 350°F. Grease two mini donut pans with butter or nonstick spray, or work in batches.
Whisk the flour, baking powder, baking soda, cinnamon, nutmeg, allspice, and salt in a large bowl.
Whisk the egg, granulated sugar, and dark brown sugar in a separate bowl until smooth. Whisk in the melted butter, reduced cider, milk, and vanilla.
Add the wet ingredients to the dry ingredients and fold gently with a spatula just until combined. Do not overmix.
Transfer the batter to a piping bag or zip-top bag with the corner snipped off. Pipe the batter into the donut cavities, filling each about halfway.
Bake for 8 to 10 minutes, until the donuts spring back lightly when touched and a toothpick inserted into the thickest part comes out clean. Cool in the pans for 5 minutes, then transfer to a wire rack.
Combine the granulated sugar, cinnamon, and nutmeg in a shallow bowl.
While the donuts are still slightly warm, dip both sides in the salted melted butter, then generously coat with the cinnamon sugar.
Serve warm or at room temperature. Refrigerate leftovers for up to 3 days.