Combine starter with milk:
In a large bowl, combine the sourdough starter with milk, whisking until the starter is completely incorporated.Mix wet ingredients:
Mix in the egg, sugar, melted butter, and vanilla, stirring until uniform.Sift and mix dry ingredients:
Sift the flour and salt into the bowl, then mix until the batter is smooth and free of lumps.Rest dough 30 minutes:
Cover the dough with a damp cloth and allow it to rest for 30 minutes.Knead dough 12 minutes:
Using a stand mixer fitted with a dough hook, knead on medium speed for 12 minutes until the dough cleans the sides. Alternatively, develop gluten through stretch and folds.Bulk rise overnight:
Cover the bowl with a damp cloth and let the dough rise at 70°F (20°C) for 8-10 hours, or overnight, until doubled and airy.Roll dough 1/2 inch:
Transfer the dough to a well-floured surface and roll to a thickness of 1/2 inch. If the dough resists, let it relax for 10 minutes before continuing.Cut donut shapes:
Cut out donuts using a 2 3/4-inch biscuit cutter. Collect the scraps, rest them for 30 minutes, then re-roll and cut additional donuts.Form donut holes:
Make a hole in the center of each donut by piercing with a chopstick or skewer and spinning to enlarge to 1 inch. Alternatively, use your fingers to form the hole.Proof donuts 2 hours:
Arrange the donuts 2 inches apart on a parchment-lined surface, cover with a damp cloth, and let rise at room temperature for about 2 hours until increased by 50%.Bake donuts 13-15 minutes:
Preheat the oven to 350°F (175°C). Move the parchment with donuts onto a baking sheet and bake on the center rack for 13-15 minutes.Fry donuts golden brown:
In a deep-sided frying pan, heat 1 inch of coconut oil over medium heat to 350°F (175°C). Fry donuts in batches for 1 minute per side until golden brown. Drain on paper towels.