Reduce cider: In a medium saucepan, bring the apple cider to a boil over medium-high heat. Cook, stirring occasionally, until syrupy and reduced to about 120 ml / ½ cup, 18–22 minutes. Pour into a medium bowl. Whisk in the apple butter, sour cream, and vanilla. Let cool slightly.
Mix dry ingredients: In another medium bowl, whisk together flour, baking powder, salt, baking soda, cinnamon, and nutmeg.
Cream eggs and sugars: Using a stand mixer fitted with the paddle or a hand mixer, beat the brown sugar, and eggs on medium-high until light and fluffy, 3–4 minutes. Add the melted, cooled butter and mix until incorporated, about 1 minute.
Combine wet and dry ingredients: Reduce mixer speed to low. Add the dry ingredients in two additions, alternating with the cider mixture in two additions, starting and ending with the dry. Mix just until combined. Dough will be very soft and sticky.
Shape and chill dough: Thoroughly flour a parchment-lined baking sheet with about ⅓ cup flour. Scrape dough onto it. Dust the top and your hands with more flour and gently pat into a ¾-inch thick rectangle. Lightly flour again, cover tightly with plastic wrap, and chill at least 3 hours or up to overnight.
Make cinnamon sugar: Stir together the sugar, cinnamon, nutmeg and freeze dried apple powder (if using) in a wide, shallow bowl. If your apple powder has clumped up, push it through a fine mesh sieve to remove any lumps.
Cut doughnuts: Cut 10 squares of parchment, about 4 inches each. Working directly on the floured baking sheet, dip your cutter(s) in flour and punch out rounds. If using two cutters, cut the centre holes. Place each doughnut on its own parchment square. Gather scraps (including holes), gently re-roll once, and cut more rounds—don’t overwork or doughnuts will be tough. You should get about 10 doughnuts plus the bonus (chef's snack!) holes.
Set up frying station: Place a wire rack over a paper towel–lined baking sheet. Fit a large heavy pot with a deep-fry thermometer. Add oil to a depth of 3 inches and heat over medium-high to 350°F (177°C).
Fry doughnuts: Working in batches, carefully lower donuts (parchment and all) into the hot oil; after a few seconds, peel away parchment with tongs. Fry until deep golden brown, about 3 minutes per side. Transfer to the rack. Repeat with doughnut holes, about 2 minutes per side. Return oil to 350°F between batches.
Coat & serve: While still warm, toss doughnuts and holes in the cinnamon sugar. Best enjoyed within a few hours of frying.