Simmer the apple cider in a small saucepan until reduced to about 3/4 cup. Let cool 10 minutes.
Stir the coating sugar and cinnamon in a shallow bowl. Set aside for finishing.
Whisk flour, baking powder, baking soda, cinnamon, nutmeg, and salt in a large bowl.
Whisk melted butter with granulated and brown sugars until sandy. Beat in the eggs until smooth.
Pour in the cooled reduced cider, buttermilk, and vanilla. Whisk to combine.
Fold the dry mixture into the wet just until a soft dough forms. Rest 10 minutes.
Heat oil in a heavy pot to 350°F. Maintain temperature while frying.
Lightly flour the counter. Pat or roll dough to 1/2 inch thick. Cut doughnuts and holes.
Fry in batches 1–2 minutes per side until deep golden. Drain on a rack and toss warm doughnuts in cinnamon sugar.
For an optional glaze, whisk powdered sugar with cider until smooth. Dip warm doughnuts and let set.