4 1/4 cups all-purpose flour, plus more for work surface
1 cup apple cider
1/4 cup whole milk
1/3 cup lightly packed dark brown sugar
2 large eggs
1 1/3 cups granulated sugar, divided
4 tablespoons cold unsalted butter, cut into small pieces
1 teaspoon baking powder
1 teaspoon kosher salt
1/2 teaspoon baking soda
1/2 teaspoon ground nutmeg
4 teaspoons cinnamon, divided
Canola oil, for frying
Line a baking sheet with parchment paper and dust generously with flour. Whisk the apple cider, milk, brown sugar, eggs, and 1/3 cup granulated sugar in a large bowl until combined. Pulse the butter, baking powder, salt, baking soda, nutmeg, 4 1/4 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 1/2-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.