Step 1

Bring cinnamon sticks and apple cider to a boil in a large skillet over medium-high heat and cook until liquid is thick, syrupy, and reduced to about ⅓ cup, 20–30 minutes. Scrape into a medium bowl and whisk in apple butter, buttermilk, and vanilla until incorporated; set aside.

Step 2

Whisk baking powder, salt, baking soda, nutmeg, 3½ cups flour, and 1 tsp. ground cinnamon in a medium bowl. Using an electric mixer on medium-high speed, beat butter, brown sugar, and ¼ cup granulated sugar in a large bowl until light and fluffy, about 4 minutes. Add eggs one at a time, beating until well blended after each addition. Reduce mixer speed to low and add dry ingredients in 2 additions, alternating with cider mixture in 2 additions, starting with dry ingredients and ending with cider mixture (dough will be very soft and sticky).

Step 3

Scrape dough onto a parchment lined-rimmed baking sheet thoroughly dusted with flour (about ⅓ cup). Dust hands and top of dough with more flour, then gently pat dough to ¾" thick. Dust with more flour and tightly cover with plastic wrap; chill dough at least 3 hours.

Step 4

Whisk remaining 1 cup granulated sugar and 1 Tbsp. cinnamon in a small bowl until no lumps remain.

Step 5

Working on baking sheet, punch out as many rounds as you can with 3¼" cutter, then use 1¼" cutter to punch out center of each round. Gather doughnut scraps, reserving holes, and gently re-roll without overworking dough; repeat until all dough has been used (you should have 18 doughnuts).

Step 6

Set a wire rack inside a paper towel-lined rimmed baking sheet. Fit a large pot with deep-fry thermometer and pour in oil to a depth of 3". Heat over medium-high until thermometer registers 350°. Working in batches, fry doughnuts until deep golden brown, about 3 minutes per side. Transfer to prepared rack and let cool slightly. Fry doughnut holes until deep golden brown, about 2 minutes per side. Transfer to prepared rack and let cool slightly. Toss warm doughnuts and doughnut holes in cinnamon sugar.

Step 7

Do Ahead: Dough can be made 1 day ahead; cover and chill.