Bring apple cider to a boil over medium-high heat. Reduce down to ½ cup (120 ml) (takes about 20-25 minutes).
Pour into a large glass measuring cup.
Add butter to hot cider and stir to melt. Set aside, cool.
In a large bowl, whisk together eggs with white and brown sugar. Add cooled cider/butter mixture, sour cream, and vanilla extract.
In a large bowl or in a fine mesh strainer set over the wet ingredients, sift together flour, baking powder, salt, cinnamon, and nutmeg.
Using a wooden spoon, mix dry ingredients into the wet until just incorporated.
Cover and chill for 1 hour.
On a very well-floured surface, roll out the cooled dough to about ¾-inch (2 cm) thickness and cut into donut shapes using pastry cutters or a mason jar/cup and a bottle cap/shot glass. Chill cut donuts 10 minutes more if necessary.
Heat oil to 350˚F (180˚C).
Fry donuts, 3-4 at a time, flipping occasionally until golden on both sides.
Remove to a paper towel lined baking sheet to drain. Roll in cinnamon sugar while still warm.