Pour the apple cider into a medium saucepan, place the pan on medium heat, and bring it to a low boil. Continue to cook the apple cider until it reduces down to 3/4 cup, which might take 45-55 minutes. This step can be done ahead of time and the apple cider concentrate can be stored in the refrigerator until you're ready to use it.
Divide up the 3/4 cup apple cider concentrate: Pour 1/4 cup of it into a small bowl. Pour another 1/4 cup of it into the bowl of a large stand mixer, and reserve the remaining 1/4 cup to make the glaze later.
Heat the small bowl of apple cider concentrate in the microwave in short 10-second bursts, until it is warm but not scalding—it should be around 105-110 F, so hotter than body temperature. Sprinkle the yeast on top, stir it in gently, and let it sit for about 5 minutes, until the yeast foams up.
To the large mixing bowl with the apple cider concentrate, add the milk, butter, shortening, sugar, yolks, salt, cinnamon, and nutmeg. Pour the yeast mixture into the large mixing bowl as well, then add half of the flour (7 1/3 oz, or 1 2/3 cups). Mix with the paddle attachment for 3-5 minutes, until very smooth.
Add the rest of the flour (7 1/3 oz, or 1 2/3 cups) and mix with the paddle for another 2 minutes. Switch to the dough hook and mix for 5-7 minutes, until the dough forms a soft, smooth ball around the hook and pulls away from the sides of the bowl. If it seems very sticky while it’s mixing, add a touch more flour, but only another tablespoon or two. Let the mixing do most of the work and resist the temptation to add too much flour, or else you will have tough doughnuts. When the dough is smooth, supple, and pulling away from the sides of the bowl, turn off the mixer. It will still be a bit soft and sticky!
Generously oil a large bowl and turn the doughnut dough into the oiled bowl. Cover loosely with a cloth and set it in a warm place to rise for 1 hour, until doubled in size.
To Make the Apple Filling:
While you wait for the dough to rise, prepare the apple filling. Peel, core, and chop the apples into small pieces about 1/2-3/4" wide. Place a large saucepan over medium heat, and melt the butter. Add the sugar, and stir it in until the sugar melts. (The mixture will be sandy and grainy.) Add the apples, cinnamon, vanilla, salt, and apple cider vinegar.
Cook the apples, stirring occasionally, until they are golden and caramelized, and almost all of the liquid has evaporated from the pan. The time required depends on your pan and your apple variety, but 15-20 minutes is a rough cooking estimate. Scrape the apples into a shallow pan or bowl, and refrigerate them until they're room temperature or cooler. (Apples can be made ahead of time and kept in the refrigerator until ready to use.)
To Assemble, Fry, and Finish:
After an hour, dust your work surface with flour, punch down the dough, and turn it out onto your work surface. Gently pat it or roll it into a thin layer about 1/2-inch thick. Spread half of the apples over half of the dough. Fold the dough over on itself, and press down gently to seal the apples into the middle of the dough.
Press the dough out into a thin rectangle again, flour it as necessary to keep it from sticking. Add the rest of the apples on half of the dough, and fold it over itself again. (This process helps distribute apples randomly throughout the dough layers.) Pat the dough back into a thin rectangle.
Use a bench scraper or large sharp knife to cut the dough into small squares, about 1 1/2" or so. Line two baking sheets with parchment , and dust it lightly with flour. Gather together about 4-5 squares (total weight 2.5-3 oz) and press them together into a patty, pinching them at the top to help secure them. Don't worry if the patties seem to have a ragged appearance, or if apples poke out—this assembly method produces the beautifully craggy and randomly-shaped signature apple fritter appearance! Place a fritter on a baking sheet, then repeat until all of your fritters are formed. You should get about 15 fritters from this recipe.
Cover the fritters loosely with cloths and let them rise in a warm place until nearly doubled, about 30 minutes.
Toward the end of the fritter rise time, pour the frying oil into a medium-large saucepan, and insert a deep fry thermometer. Turn the heat to medium and heat until the oil reaches 350 F. During the frying process, keep a close eye on the thermometer and monitor the temperature carefully.
Once the fritters have risen and the oil is the right temperature, gently place 2-3 fritters at a time in the oil and fry them for 1-2 minutes per side, until they are puffed and golden brown. Once cooked, remove them from the oil using a slotted spoon or frying tool, and place them on a paper-towel lined wire rack to cool. Repeat with the remaining fritters. Periodically remove any stray apples or dough that's floating in the oil. If the oil gets too hot, turn the heat down or take the pan briefly off the heat, and if it gets too cool, pause your frying until it has a chance to heat up again.
To make the glaze, whisk together the reserved 1/4 cup apple cider concentrate along with the rest of the ingredients. Try brushing some on a warm (but not super hot!) fritter. It should brush easily, and start to melt into the outside of the fritter. If it is too thick, and looks more like frosting than glaze, add a little apple cider or milk, a small spoonful at a time, to get the right consistency. Brush the glaze on the fritters, or spoon it on and spread it around with the spoon. Repeat until all of the fritters have been glazed.
Let the glaze set for about 20 minutes, then serve! These fritters are best on the day they are made, but they will keep for several days in an airtight container at room temperature.