In a large bowl, combine the active sourdough starter and granulated sugar. Whisk until the sugar is dissolved.

In a separate bowl, whisk together 1 cup of the all-purpose flour, salt, and warmed milk. Let it sit for 5 minutes.

Add the sourdough mixture, the flour and milk mixture, the egg, and melted butter to the bowl of a stand mixer fitted with a dough hook. Mix on low speed until just combined.

Gradually add the remaining 1.5 cups of flour, mixing on low speed until a shaggy dough forms. Increase speed to medium-low and knead for 6-8 minutes, until the dough is smooth and elastic.

Transfer the dough to a lightly greased bowl, cover with plastic wrap, and let it rest at room temperature for 4-6 hours, or until doubled in size.

Punch down the dough. Turn it out onto a lightly floured surface and gently roll it to about 0.5 inch thickness. Cut out donuts using a donut cutter or two round cutters of different sizes.

Place the cut donuts on parchment-lined baking sheets. Cover loosely with plastic wrap and let them proof at room temperature for 1-2 hours, until puffy.

Heat the vegetable oil in a large, heavy-bottomed pot or Dutch oven to 350°F (175°C).

Carefully fry the donuts, a few at a time, for 2-3 minutes per side, until golden brown. Use a slotted spoon to remove them from the oil.

Transfer the fried donuts to a wire rack set over paper towels to drain.

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

Dip each warm donut into the glaze, letting excess drip off. Place back on the wire rack to set.