You will also need a candy/fat thermometer for frying the donuts, as well as a 12- to 18-inch pastry bag fitted with a 1/4-inch round tip to fill them.
Game plan: The dough can be made in a stand mixer fitted with a hook attachment if you prefer. Place the yeast and sugar in the stand mixer’s bowl and proceed with the recipe, mixing at medium speed until the dough comes together and forms a ball that is smooth, shiny, and elastic, about 5 minutes.
When deep-frying the donuts, make sure the oil stays at a constant temperature, adjusting your stove’s heat as necessary.
1Place 2 1/2 cups of flour, the cinnamon, salt, and nutmeg in a large bowl and whisk to aerate and combine; set aside. Coat a second large bowl with vegetable oil; set aside.
2Place the yeast and 1 teaspoon of the sugar in a medium bowl. Heat 1/2 cup of the apple cider until warm, between 105°F and 115°F. Add to the yeast and sugar and stir to combine. Let sit until the mixture is foaming, about 5 minutes.
3Add the remaining 1/4 cup sugar, remaining 1/4 cup apple cider, egg yolks, apple butter, and vanilla to the yeast mixture and whisk to combine. Add this mixture to the reserved flour mixture and stir with a wooden spoon until the dough comes together and begins to form a ball.
4Transfer the dough to a lightly floured work surface. Scatter the butter pieces over the dough and knead until the butter is fully incorporated and the dough is smooth, shiny, and elastic, about 6 to 8 minutes. Add up to 1/4 cup of additional flour as needed if the dough is sticky. Form the dough into a ball, place it in the oiled bowl, and turn to coat it in the oil. Cover the bowl with plastic wrap or a damp towel and let the dough rise in a warm place until doubled in size, about 2 hours. Meanwhile, make the filling.
For the caramel filling:
1Combine the sugar, water, and salt in a medium saucepan and bring to a boil over medium heat, stirring often until the sugar has completely dissolved. Continue to boil, swirling the pan occasionally (but not stirring), until the mixture turns a deep amber color, about 10 to 15 minutes.
2Remove the pan from the heat, carefully pour in the cream and vanilla (the mixture will bubble up and steam), and stir until evenly combined. Transfer to a medium heatproof bowl and let the caramel cool at room temperature, uncovered, for about 30 minutes before using.
1Line a baking sheet with parchment paper; set aside. Punch down the dough, transfer it to a lightly floured work surface, and roll it out until it’s about 1/4 inch thick. Using a 2-1/2-inch round cutter, stamp out as many dough rounds as possible and place them on the prepared baking sheet about 1/4 inch apart. Gather the dough scraps into a ball and roll out and cut again. Discard any remaining dough scraps.
2Cover the dough rounds loosely with plastic wrap or a damp towel. Let rise in a warm place until puffy and about 1/2 inch thick, about 30 minutes.
3Heat the oil in a Dutch oven or a large, heavy-bottomed pot over medium heat until the temperature reaches 365°F on a candy/fat thermometer. Meanwhile, fit a wire rack over a second baking sheet; set aside. Place the sugar in a large bowl; set aside. Transfer the cooled caramel to a piping bag fitted with a 1/4-inch round tip; set aside.
4When the oil is ready, add 4 of the dough rounds and fry until golden brown, flipping halfway through, about 2 minutes total. (If air bubbles appear in the donuts, pierce them with the tip of a paring knife.) Remove the donuts with a slotted spoon to the rack. Add 4 more dough rounds to the oil. While these dough rounds are frying, use tongs to transfer the first 4 (still-hot) donuts to the bowl of sugar. Toss to coat in the sugar, then return to the wire rack. Repeat frying and sugarcoating the remaining dough rounds.
5When the donuts are cool enough to handle, use a paring knife to puncture the side of each to form a pocket in the center. Place the tip of the piping bag into the pocket and pipe about 1 heaping teaspoon of caramel inside. Serve warm or at room temperature.