Bring cinnamon sticks and cider to a boil in a large skillet over medium high heat, cook until liquid is thick and syrupy, and reduced to ⅓ cup (about 20-30 minutes) Reserve 1 tbsp of liquid for glaze.

In a medium bowl whisk together 1 egg, vanilla, buttermilk, and ⅓ cup reduced cider until well incorporated.

In a separate medium bowl, whisk together 3.5 cups flour, baking powder, salt, baking soda, nutmeg, and 1 tsp ground cinnamon.

In a large bowl using an electric mixer, beat butter, brown sugar, and ¼ cup granulated sugar until lighter in color and fluffy (about 4 minutes) Add 2 eggs one at a time, blending well after each addition. Reduce mixer speed and add dry ingredients in two additions, and the wet ingredients in two additions alternating-- dry, wet, dry, wet. Dough will be very soft and sticky.

Dust a parchment lined baking sheet with ⅓ cup of flour. Turn out the dough onto the surface. Top dough with more flour and dust hands, then gently pat to ¾” thick. Dust with more flour and cover with plastic wrap. Chill at least 3 hours.

Mix together all glaze ingredients, stir until smooth. Add more milk or cream to reach desired consistency. Set aside.

Working on the baking sheet, punch out as many rounds as you can. For full size donuts, use a 3 1/4” cutter, and use a 1 ¼” cutter to punch out the centers. Leftover dough can be patted out once more and re cut. Avoid patting out more than once, as the dough can become tough.

Fry! Fit a wire rack inside a baking sheet lined with paper towels. In a medium bowl, whisk together 1 cup sugar and 1 tbsp cinnamon and set aside. In a large pot, pour 3” of oil. Heat over medium high until the thermometer registers 350F. Working in batches, fry donuts until deep golden brown, about 1-2 minutes per side for donut holes, 3 minutes per side for full size donuts. Transfer to prepared rack and cool slightly. Toss warm donuts in cinnamon sugar. Drizzle the glaze on top, or serve on the side for dipping.