Step 1: Prepare the Dough
Activate the yeast: In a small bowl, combine warm milk and pinch of sugar. Sprinkle the yeast over the top and let it sit for 5-10 minutes until it becomes foamy. (The yeast will help your donuts be more fluffy and not dense.)
Mix wet ingredients: In the bowl of a stand mixer with a dough hook, mix together the sourdough discard starter, warm milk, sugar, eggs, softened butter, and vanilla extract until smooth.
Combine dry ingredients: In another bowl, whisk together the whole wheat flour, and salt.
Mix the dough: Gradually add the dry ingredients to the wet ingredients, mixing until just combined. The dough will be slightly sticky but should hold its shape. Turn the mixer on medium and knead for about 5 minutes until the dough comes together. At this point it should be slightly sticky and peeling off the bottom and sides of the bowl. If it's too dry, not sticky at all, add a little warm water (1 tablespoon at a time) until you reach a dough that is smooth, not too wet, yet slightly sticky. Then, place the dough in a greased bowl. Likewise, if the dough is too wet, add a tablespoon of flour at a time until it comes together and has a slight stickiness.
Let it rest: Cover the dough with a damp tea towel or plastic wrap and let it rest for at least an hour, or until it has doubled in size. At this point you can do a longer fermentation, and place the dough in the fridge overnight. This will lead to a stronger sourdough flavor.
Step 2: Shape the Donuts
Roll out the dough: Lightly flour your work surface and roll the dough to about ½-inch thick. If the dough is too sticky, sprinkle some flour on top.
Cut the donuts: Use a donut cutter or two round cookie cutters (one large, one small) to cut out the donuts and donut holes, and place on a piece of parchment paper. Keep a good distance between the donuts so that they don't expand into each other. Place a damp towel over them.
Rest again: Let the shaped dough rest for 1 hour in a warm space.
Step 3: Cook the Donuts
Prepare to fry: Fill a deep fryer, large pot, or Dutch oven with coconut oil, ensuring it's about 2-3 inches deep.
Preheat the deep frying oil: Preheat the coconut oil in a deep fryer, large pot, or dutch oven to 350°F. (For best results see TIPS.)
For frying: Gently lower the donuts into the hot oil, cooking for 1-2 minutes per side until golden brown. Use a slotted spoon to remove them from the oil and place them on a paper towel-lined plate to drain.
Step 4: Glaze
Make the glaze: In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. Add more milk to adjust the thickness if needed.
Glaze the donuts: Once the donuts are slightly cooled, dip the cooked donuts into the glaze, flipping them over so that the glaze coats both sides, allowing excess to drip off.
Step 5: Serve
Let the donuts rest for a few minutes to allow the glaze to set, then enjoy! These donuts are best served fresh but can be stored in an airtight container for a couple of days.