In a large mixing bowl, whisk together the sourdough discard, granulated sugar, egg, warm milk, melted butter, and vanilla extract until well combined.

In a separate bowl, whisk together the all-purpose flour, baking powder, and salt.

Gradually add the dry ingredients to the wet ingredients, mixing with a wooden spoon or spatula until a shaggy dough forms.

Turn the dough out onto a lightly floured surface and knead for 2-3 minutes until it comes together and is mostly smooth. Avoid over-kneading. The dough will be slightly sticky.

Place the dough back into the bowl, cover with plastic wrap, and refrigerate for at least 1 hour, or up to 2 hours. This chilling step is crucial for making the dough easier to handle and preventing it from absorbing too much oil during frying.

On a lightly floured surface, roll out the chilled dough to about 1/2 inch thick. Use a donut cutter (or a 3-inch round cutter for the donut and a 1-inch cutter for the hole) to cut out the donuts and donut holes. Gather the scraps, re-roll once, and cut out more donuts.

Line a baking sheet with parchment paper. Carefully transfer the cut donuts and holes to the prepared baking sheet. Let them sit at room temperature for 15-20 minutes while you heat the oil.

In a deep pot or Dutch oven, pour enough vegetable oil to reach about 3 inches deep. Heat the oil over medium heat until it reaches 350°F (175°C) on a deep-fry thermometer. Maintain this temperature throughout frying.

Carefully drop 2-3 donuts (or more, depending on your pot size) into the hot oil, being careful not to overcrowd the pot. Fry for 1-2 minutes per side, until golden brown. Donut holes will fry faster, about 30-60 seconds per side.

Using a slotted spoon or spider skimmer, remove the fried donuts and donut holes and place them on a wire rack set over paper towels to drain excess oil.

While the donuts are still warm, prepare the glaze. In a medium bowl, whisk together the sifted powdered sugar, milk (start with 3 tbsp and add more if needed for desired consistency), and vanilla extract until smooth.

Dip each warm donut into the glaze, turning to coat both sides. Place them back on the wire rack (without paper towels underneath) to allow the excess glaze to drip off and set. Glaze donut holes similarly.

Serve the Delicious Sourdough Discard Donuts warm and enjoy!