Preheat oven to 375° F with a rack in the center. Spray two donut pans lightly with cooking spray.
Pour the cider into a frying pan and set over high heat. Boil until reduced to ½ cup, which takes about 10 minutes. Set aside to cool. (If you over-reduce it by accident, just top it up with regular cider. Not that I've ever done that, ahem.)
In a medium bowl, whisk together the all-purpose and whole-wheat flours, regular and brown sugars, baking soda, baking powder, cinnamon, nutmeg and salt.
In a small bowl, whisk together the egg, buttermilk, butter and reduced apple cider.
Pour the wet ingredients into the dry and stir just until combined. You don't want any pockets of flour remaining, but you don't want to over-mix.
You can spoon the batter into your prepared donut pans, but I prefer to pour the batter into a gallon-sized zip top plastic bag, snip off a small piece of one of the lower corners, and use it like a pastry bag to fill each well about ⅔ full.
Bake until the donuts spring back a bit when touched lightly, about eight minutes. Leave to cool for a couple of minutes, then gently remove donuts from pan. Cool on a rack.
For the topping, melt butter and pour into a wide, shallow bowl. In a separate bowl, mix together sugar and cinnamon. Dip the top of each donut into the butter for just a second and then into the cinnamon sugar to coat as much as you like. Serve warm or at room temperature.