Step 1

In a small bowl combine the yeast and warm cider; set aside until it foams up, about 5 minutes.

Step 2

In the bowl of a stand mixer fitted with the hook attachment, combine the melted butter and sugar, and - using a silicon whisk - mix until well combined. Whisk in the egg, egg yolk, milk, salt, nutmeg, cinnamon, yeast mixture, and 3 cups of flour, and beat on medium-low speed until smooth. Beat in remaining flour just until a soft ball of dough forms around the hook. Don't knead the dough yet!

Step 3

Transfer the dough to a large, lightly greased bowl, turning once to coat the top. Cover tightly with saran wrap and set aside to rise in a warm area until it appears to have tripled in size, about 1 hour.

Step 4

Gently punch the dough down and turn it out onto a lightly floured work surface. Knead the dough into a smooth ball, about 7-8 times, or until it's uniform in texture.

Step 5

Using a rolling pin, roll the dough out into a large rectangle, approximately 1/2 inch thick. Using a lightly floured round dough cutter, cut an equal amount of circles from the rectangle.

Step 6

Transfer the cut doughnuts to a large, lightly greased baking sheet, spacing them an inch a part (they will rise). Cover pan lightly with saran wrap and set aside for 45 minutes to an hour, or until the doughnuts have almost doubled in size.

Step 7

When ready to fry, heat oil in a large, heavy bottomed pot, to 360°(F). Fry doughnuts, a few at a time, for 1-2 minutes on each side, or until golden brown.

Step 8

Quickly roll the warm doughnuts in the cinnamon sugar and set aside.

Step 9

Peel, core, and dice the apples. Set aside.

Step 10

In a large skillet over medium-heat, melt the butter and continue cooking until it's a light golden color; about 3 minutes. Add in the apples, salt, and both sugars. Cook, stirring occasionally, until apples are completely softened, about 10 minutes. Add in the cinnamon and apple cider and continue cooking until apples are soft and sticky, 1 to 2 more minutes. Stir in flour, mix well, and allow mixture to thicken; about 30 seconds. Remove from heat and set aside until needed.

Step 11

In a medium-sized shallow bowl, combine the granulated sugar and cinnamon and set aside until needed.

Step 12

Scrape the apple pie filling into a pastry bag fitted with a round tip and set aside.

Step 13

Using the back of a small spoon (or your finger) make a small hole on the side of each doughnut.

Step 14

Carefully fit the nozzle of the bag into a doughnut and squeeze until you see the center begin to puff up. Repeat with all doughnuts.

Step 15

Donuts are best served the day they are made, but will keep for 24 hours.