Add apple cider to a small saucepan over medium-high heat. Simmer and reduce the cider to ¾ cup, about 30 to 35 minutes. Pour into a medium mixing bowl and let cool completely.

Whisk the melted butter, buttermilk, and eggs into the cooled cider.

In a large mixing bowl, whisk together the flour, brown sugar, ½ cup of the granulated sugar, 1 tablespoon of the cinnamon, baking powder, salt, baking soda, and nutmeg until combined.

Add the wet ingredients into the dry, mixing on low speed until a slightly tacky dough forms, adding additional flour if needed. Cover with plastic wrap and chill until the dough is firm, at least 1 hour and up to 12 hours.

Fill a Dutch oven with 3 inches of oil. Heat the oil to 350ºF, letting the oil come back to heat between batches.

Turn the chilled dough out onto a generously floured, clean work surface. Flour the top of the dough and pat the dough into a ½ inch thick rectangle.

Dip a 3-inch round cookie cutter in flour and stamp out donuts. Use a 1-inch cutter to cut the holes in the center. Add each donut to individually cut squares of parchment paper.

Working a few at a time, add the donuts with the parchment paper to the hot oil, removing the parchment paper with tongs as soon as it separates from the donut. Fry until golden brown on both sides, flipping halfway through cooking, about 3 minutes total for the large donuts and about 2 minutes total for the donut holes.

Use tongs or a slotted spoon to transfer to a wire cooling rack for about 5 minutes.

In another shallow bowl, stir together the remaining granulated sugar and the remaining cinnamon. Toss the donuts in the cinnamon sugar and serve.