Reduce the apple cider: In a small saucepan, bring apple cider to a gentle boil and let simmer over medium heat until thick and reduced to about 1/3 cup (80 ml), 15-25 minutes. Set aside to cool.
Preheat oven to 350°F/180°C. Grease donut pans with butter or nonstick spray. Set aside.
In a small bowl whisk together the egg, granulated sugar and brown sugar, until well combined. Whisk in the melted butter. Add the reduced apple cider, buttermilk and vanilla extract, and mix until combined. Pour the mixture into the flour mixture and whisk just until combined and smooth. Don't overmix the batter.
Divide the mixture between the donut cavities, filling about halfway full. I recommend using a piping bag to pipe the batter, it will be much easier and less messy. Bake for about 10 minutes until a toothpick inserted into the center of the donuts comes out clean. Let donuts cool in the pan for 5 minutes and then transfer them to a wire rack.