• In a large mixing bowl, combine the active sourdough starter, warm milk, and sugar. Mix well.
• Add the eggs, melted butter, and vanilla extract. Stir until combined.
• Gradually add the flour and salt, mixing until a soft dough forms. Knead the dough until it becomes smooth and elastic, about 5-7 minutes. If using a stand mixer, use the dough hook attachment.
• Place the dough in a lightly greased bowl, cover it with a damp cloth or plastic wrap, and let it rise in a warm place for 4-6 hours, or until it has doubled in size.

Shape the Donuts:

• On a floured surface, roll out the dough to about 1/2-inch thickness. Use a donut cutter or two round cutters (one larger and one smaller) to cut out donut shapes. Gather the scraps, re-roll them, and cut out more donuts as needed.
• Place the cut donuts on a parchment-lined baking sheet, cover them with a cloth, and let them rise for an additional 1-2 hours, or until they are puffy.

Fry the Donuts:

• In a deep fryer or a large, heavy-bottomed pot, heat the vegetable oil to 350°F (175°C). Carefully fry the donuts in batches, turning them once, until they are golden brown on both sides (about 1-2 minutes per side).
• Use a slotted spoon to transfer the fried donuts to a paper towel-lined plate to drain excess oil.

Prepare the Glaze:

• In a medium bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.

Glaze the Donuts:

• While the donuts are still warm, dip them in the glaze, allowing any excess to drip off. Place the glazed donuts on a wire rack to set.