Whisk the flour, baking powder, 1 1/2 teaspoons of the cinnamon, salt, baking soda and nutmeg in a medium bowl.
Beat 2/3 cup of the sugar and shortening in another bowl with a mixer on medium speed until sandy. Beat in the egg and yolk, then gradually mix in the applesauce, scraping the bowl. Beat in half of the flour mixture, then the buttermilk and vanilla, and then the remaining flour mixture. Mix to make a sticky dough; do not overmix.
Scrape the dough onto a lightly floured sheet of parchment paper in a baking sheet and pat into the size of the baking sheet, about 1/2-inch thick. Cover with plastic wrap and refrigerate at least 2 hours or overnight.
Heat 2 inches of oil in a large heavy-bottomed pot over medium-high heat until a deep-fry thermometer registers 350 degrees F. Line a baking sheet with paper towels. Cut the chilled dough into 12 rounds, using a floured 3-inch biscuit cutter, then cut out the middles with a 1-inch cutter (or use a doughnut cutter). Slip 2 or 3 doughnuts at a time into the hot oil and fry until golden brown, 1 to 2 minutes per side, adjusting the heat as needed. Transfer to the paper towels to drain.
Mix the remaining 1 cup sugar and 1 tablespoon cinnamon in a medium bowl. Toss the doughnuts in the cinnamon-sugar. Serve warm.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)