1. Combine Wet Ingredients for the Overnight Rise: In a large mixing bowl, whisk together your active sourdough starter, warm milk, 1/4 cup granulated sugar, egg, and vanilla extract until well combined and smooth. This creates the flavorful base for your donuts.

2. Add Dry Ingredients and Form the Dough: Gradually add the all-purpose flour and salt to the wet mixture. Mix with a wooden spoon or your hands until a shaggy dough forms. It doesn’t need to be perfectly smooth at this stage.

3. Incorporate Butter and Knead Lightly: Pour the melted and cooled butter over the dough. Knead the dough gently in the bowl for about 2-3 minutes until the butter is mostly incorporated and the dough comes together. It will still be a bit sticky.

4. First Rise (Overnight): Cover the bowl tightly with plastic wrap or a damp kitchen towel. Place it in the refrigerator for 8-12 hours, or overnight. This slow, cold fermentation develops the sourdough flavor and texture. The dough should look slightly puffed and bubbly in the morning.

5. Prepare for Frying and Second Rise: 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 different-sized round cutters (one larger for the donut, one smaller for the hole) to cut out your donuts. Place the cut donuts and donut holes on a floured baking sheet.

6. Second Rise (Room Temperature): Cover the cut donuts loosely with plastic wrap or a clean kitchen towel. Let them rise at room temperature for 30-60 minutes, or until they look visibly puffy and light. They should spring back slowly when gently poked.

7. Heat Oil and Fry the Donuts: While the donuts are rising, heat about 2-3 inches of vegetable oil in a large, heavy-bottomed pot or Dutch oven over medium heat to 350-375 F (175-190 C). Use a thermometer to ensure the oil is at the correct temperature. Carefully lower 2-3 donuts into the hot oil, being careful not to overcrowd the pot.

8. Cook Until Golden Brown: Fry the donuts for 1-2 minutes per side, or until they are golden brown and puffed. Use a slotted spoon or spider to carefully flip them. Once cooked, transfer the donuts to a wire rack lined with paper towels to drain excess oil. Immediately toss them in extra granulated sugar or your desired topping while still warm. Repeat with remaining donuts and donut holes.