In a small saucepan, simmer 2 cups of fresh apple cider or homemade apple cider over low heat until it's reduced to about ½ cup. This takes around 30 minutes. Cool before using.
Activate the yeast: In a small bowl, combine the warm milk with a pinch of sugar. Sprinkle the yeast on top and let it sit for 5–10 minutes until it turns foamy. (This helps make your donuts light and fluffy rather than dense.)
Mix the wet ingredients: In a large bowl of a stand mixer fitted with a dough hook, combine the sourdough discard, apple cider, warm milk with activated yeast, sugar, brown sugar, eggs, softened butter, and vanilla extract. Mix until smooth.
In a separate medium bowl, stir together the flour, cinnamon, nutmeg, allspice, pinch of clove, and salt.
Gradually add the dry ingredients to the wet mixture. Mix on low until just combined. The dough should be slightly sticky but holding its shape.
Turn the mixer to medium and knead for about 5 minutes. The dough should pull away from the sides and bottom of the bowl while still feeling tacky. (See Notes) Transfer the dough to a greased bowl. Cover with plastic wrap or a damp towel and let it rise in a warm spot until doubled in size, about 1 hour.
Optional: For deeper sourdough flavor, cover and refrigerate the dough overnight for a long, slow fermentation.
On a floured surface, roll the dough out, or pat out with hands to about ½ inch thick. If it sticks, dust the top with more flour. Use a donut cutter (or two round cookie cutters—a larger one and a smaller one for the donut holes) to shape your donuts. Place them on parchment paper, spaced apart so they have room to puff up.
Cover the shaped donuts with a damp towel and let them rest for about 1 hour in a warm spot.
Add about 2–3 inches of coconut oil to a heavy-bottomed pot, Dutch oven, or deep fryer. Heat it to 350°F. (See TIPS for best results.) Carefully lower each donut into the hot oil. Fry for 1–2 minutes per side, or until golden brown. Use a slotted spoon to transfer them to a paper towel-lined tray to drain.
Mix the granulated sugar and cinnamon together in a small bowl to make the coating. While the donuts are still warm, toss them in the cinnamon-sugar mixture one at a time until evenly coated and place on a wire rack. Serve and enjoy!