Preparation

Make the apple cider concentrate

• Put the cider and grated apple in a medium saucepan and bring to a boil over medium-high heat, stirring occasionally. Reduce the heat to low and cook, stirring occasionally, until the apple pieces are very tender and translucent and the mixture is reduced to just a smidge over 1 cup, 18 to 20 minutes. Slide the pan off the heat, strain off 1⁄4 cup (60 ml) of liquid, and set both mixtures aside separately to cool completely.

Make the donuts

• Line a cookie sheet with parchment and generously coat with flour.
• Whisk the flour, brown sugar, baking powder, salt, baking soda, cinnamon, and nutmeg in a large bowl until well blended. Put the buttermilk and eggs in a small bowl or 2-cup (480 ml) measure and whisk until well blended. Pour the egg mixture along with the melted butter and apple concentrate over the flour and, using a silicone spatula, stir and fold until it forms a moist dough (I like to use one hand to help mix while keeping the other working the spatula).
• Arrange a piece of parchment on your work surface, generously coat with flour, and put the dough in the center. Using floured hands and starting in the center and moving outward toward the edges, gently pat and shape the dough into an even 2⁄3-inch (17 mm) thick round, flouring your hands as needed. Dust the top of the dough with flour, cover with another piece of parchment, and flip the dough.
• Peel away the top piece of parchment. Using a 3-inch (7.5 cm) cookie cutter, cut out rounds, dipping the cutter in flour as needed. Using a 1-1⁄8-inch (2.9 cm) plain round cookie cutter, cut out the centers of the rounds and remove. Arrange the rounds and centers, using a floured offset spatula if needed, on the prepared cookie sheet. Stack the scraps and gently press together. Re-roll as directed, chill the dough for about 10 minutes, and cut.

Make the glaze

• Put the confectioners’ sugar and 3 Tbs. of the apple cider concentrate in a medium bowl and stir with a spoon. Add more cider concentrate, a little at a time, until the glaze is smooth, very thick, and shiny.

Cook the donuts

• Have ready a large rimmed baking sheet lined with several layers of paper towels and a Chinese-style skimmer (spider) or slotted spoon. Pour enough oil into a large pot (I use a 4-quart/950 ml Dutch oven) to fill it to a 2-1⁄2-inch (6 cm) depth and clip a candy thermometer to the side of the pan, making sure it doesn’t touch the bottom. Heat the oil to 360°F (182°C) over high heat.
• Using a long, wide spatula, carefully slip a few donuts and donut holes (don’t fry too many at one time or the temperature of the oil will drop and the donuts will be greasy) into the oil and cook, turning once with the skimmer, until deep golden brown, 65 to 70 seconds per side. With a skimmer or slotted spoon, remove the donuts from the pan, drain on the paper towel–lined baking sheet. Keep an eye on the oil temperature and adjust the heat up or down accordingly while cooking, making sure to bring the temperature back up to 360°F (182°C) before cooking subsequent batches.
• When cool enough to handle, arrange the donuts rounded side up and, using a small offset spatula, spread the tops with some of the cider glaze and dip and roll the donut holes to coat completely. Repeat with the remaining donuts. Serve warm.

Make Ahead Tips

The cider concentrate can be prepared, cooled, covered, and refrigerated for up to 3 days ahead. Bring to room temperature before using.

The dry ingredients for the dough can be prepared, covered, and stowed at room temperature for up to 3 days.