How to Make It
Line a baking sheet with parchment paper and dust generously with flour. Whisk the apple cider, milk, brown sugar, eggs, and ⅓ cup granulated sugar in a large bowl until combined. Pulse the butter, baking powder, salt, baking soda, nutmeg, 4¼ cups flour, and 2 teaspoons cinnamon in a food processor until sandy. Mix the wet ingredients into the dry ingredients with a wooden spoon until just combined (dough will be very sticky).
With floured hands, transfer the dough to the prepared baking sheet. Gently pat into a ½-inch thick rectangle, and dust with more flour. Refrigerate for 1 hour (this will make the dough easier to work with).
Meanwhile, combine the remaining 1 cup granulated sugar and 2 teaspoons cinnamon in 2 wide, shallow bowls or pie plates.
Fill a medium heavy-bottomed pot with 2 inches of oil and fit with a deep fry or candy thermometer; set over medium heat. Cut out donuts using a 3-inch donut cutter. Gather scraps, gently pat out again, and repeat until all the dough has been used.
When thermometer registers 360°F, working in batches, add donuts to the oil, being careful not to overcrowd the pot. Fry, turning once, until donuts are puffy and deep golden brown, about 2 minutes per side. Using tongs or a slotted spoon, transfer donuts to the cinnamon-sugar mixture and toss to coat. Repeat with remaining donuts, holes, and cinnamon-sugar mixture.