1. Prepare the Dough (Evening): In a large mixing bowl, combine the active sourdough starter, warm milk, granulated sugar, egg, melted butter, and vanilla extract. Whisk these wet ingredients together until well combined. The mixture should look slightly milky and uniform.

2. Add Dry Ingredients: Gradually add the all-purpose flour and salt to the wet ingredients. Mix with a wooden spoon or your hands until a shaggy dough forms. It will be sticky at this stage.

3. Knead the Dough: Turn the dough out onto a lightly floured surface and knead for about 5-7 minutes until it becomes smooth and elastic. If using a stand mixer, knead with the dough hook for 4-5 minutes. The dough should spring back slightly when poked.

4. First Rise (Overnight): Place the kneaded dough into a lightly oiled bowl, turning it once to coat. Cover the bowl tightly with plastic wrap or a damp kitchen towel. Refrigerate for 8-12 hours, or overnight. This long, cold fermentation is key for flavor development. The dough should look visibly larger and bubbly in the morning.

5. Shape the Donuts (Morning): The next morning, remove the dough from the refrigerator. On a lightly floured surface, gently roll out the dough to about 1/2-inch thickness. Use a donut cutter or two round cutters (one larger for the donut, one smaller for the hole) to cut out your donut shapes. Place the cut donuts and donut holes on parchment-lined baking sheets.

6. Second Rise: Loosely cover the shaped donuts with plastic wrap or a clean kitchen towel and let them rise in a warm spot for 1-2 hours, or until they look visibly puffy and light. They should feel airy when gently touched.

7. Fry the Donuts: Heat 2-3 inches of vegetable or canola oil in a large, heavy-bottomed pot or Dutch oven to 350 F (175 C). Use a candy thermometer to monitor the temperature; maintaining it is crucial for perfectly cooked donuts.

8. Cook and Finish: Carefully lower 2-3 donuts into the hot oil, being careful not to overcrowd the pot. Fry for 1-2 minutes per side, until golden brown and cooked through. Use a slotted spoon or spider to remove the donuts from the oil and place them on a wire rack lined with paper towels to drain excess oil. While still warm, toss them in granulated sugar or your preferred glaze. Repeat with remaining donuts and holes.