Make doughnuts: In a medium bowl, whisk together 3 3/4 cups flour, the baking powder, baking soda, salt, cinnamon, and nutmeg. In a small bowl, combine buttermilk, cider reduction, and vanilla. Using a mixer, beat granulated sugar and butter in a large bowl on medium-high speed until fluffy and very light in color, 4 to 6 minutes. "You're beating air into the butter and sugar so you have a light, delicate texture," says Davis. Add eggs one at a time, beating well after each addition and scraping bowl occasionally.
On low speed, beat in dry ingredients in 3 additions, alternating with buttermilk mixture and beating just until combined.
Line a baking sheet with parchment paper and dust generously with flour. Put dough on sheet and sprinkle generously with flour. Pat or roll dough into a slab about 1/2 in. thick. Cover with plastic wrap and chill until center feels firm enough to cut, about 1 hour.
Slip dough from parchment onto a flour-dusted work surface. Line baking sheet with a fresh piece of parchment. Cut doughnuts and holes, dipping a 2 3/4-in. doughnut cutter into flour each time and "releasing them with a strong flick of your wrist." Set on baking sheet. Gather scraps and reroll.
Heat 1 1/2 in. oil in a dutch oven to between 350° and 360° on a deep-fry thermometer. Meanwhile, set 2 cooling racks over empty baking sheets; set them near stove. Make glaze: In a medium bowl, whisk together cider reduction, powdered sugar, and vanilla.
Fry 2 or 3 doughnuts and holes at a time until a rich toasty brown, turning once with a slotted spatula or spoon, 1 to 1 1/2 minutes per side. Adjust heat as needed; "you want a lively bubble." Test the first doughnut and make sure it's cooked through (it will be light colored inside but shouldn't look wet).
Transfer doughnuts to racks over baking sheets. Let cool just until comfortably warm; at this temperature, rather than cool, "they suck up the glaze." Dip all over in glaze and set on racks.