Preheat oven to 350F. Prep your pans by greasing with butter or cooking spray, then flouring.
Cream the butter and sugars until fluffy. I do this in my stand mixer, you can also use electric beaters.
Add the eggs, one at a time, letting the first get incorporated before adding the second. Scrape the bowl as necessary.
Beat in the vanilla, cinnamon, nutmeg, salt, and baking powder.
With the mixer on low, add in the flour. Then slowly add the cider. Scrape down the sides and finish with a silicone spoon or spatula to get everything incorporated (especially the bottom of the bowl.) Don't over beat at this point.
Fill the twelve cavities of your doughnut pans. I like to spoon the batter into a large piping bag to do this neatly. Just snip off about an inch at the bottom and pipe into your pan. You should have just enough batter to make 12 doughnuts, so fill them about 3/4 full. Note: if you only have one pan just do this in batches, no worries.
Bake for 15 minutes, or just until the surfaces are firm and a toothpick comes out without wet batter. You do not want to over bake these, so make sure your oven is at the correct temperature and watch carefully. They won't be very browned on top.
Let cool in the pans for 5 minutes, then gently loosen the edges and flip out of the pan. Give the edge of the pan a rap if necessary. If the doughnuts don't fall out, let them cool a minute or two more and try again. Note: don't worry if your 'holes' have filled in, you can just poke through with a small knife, or your pinky finger, to open them up.
Whisk the sugar and cinnamon together in a bowl. Brush the doughnuts all over with melted butter, then toss in the cinnamon sugar, being sure to get all the surfaces nicely coated. Work with one doughnut at a time.
The doughnuts can be devoured asap, or saved for later. Mine were perfect the next morning. I saved mine under my glass cake dome. You can also cover loosely with foil.