In a mixing bowl, combine the sourdough discard, melted butter, milk, sugar, egg, and vanilla extract. Mix until well combined.
In a separate bowl, whisk together the flour, salt, baking soda, and baking powder.
Gradually add the dry ingredients to the wet ingredients and mix until a dough forms. You may need to use your hands to knead the dough gently until it comes together.
Place the dough on a lightly floured surface and roll it out to about 1/2-inch thickness. Use a doughnut cutter or a round cookie cutter to cut out doughnut shapes. You can use a smaller cutter to remove the centers if desired, creating classic doughnut shapes.
Place the cut doughnuts on a baking sheet lined with parchment paper and let them rest for about 15-20 minutes while you heat the oil.
Heat about 2 inches of vegetable oil in a deep pot or a deep fryer to 350°F (175°C). Use a thermometer to ensure the oil reaches and maintains the correct temperature.
Carefully place the doughnuts into the hot oil using a slotted spoon or a spider strainer, frying them in batches to avoid overcrowding. Fry for about 2-3 minutes on each side, or until they are golden brown and puffed up. Remove them from the oil and place them on a paper towel-lined plate to drain any excess oil.
While the doughnuts are cooling slightly, prepare the glaze. In a bowl, whisk together the powdered sugar, milk or water, and vanilla extract until you have a smooth glaze. Adjust the thickness by adding more liquid if needed.
Dip each doughnut into the glaze while they are still warm, then place them on a wire rack to allow any excess glaze to drip off.
Let the glaze set for a few minutes before serving. Enjoy your homemade sourdough discard doughnuts!
Notes
Feel free to get creative with toppings, such as sprinkles, chopped nuts, or even a dusting of cinnamon sugar, to customize your doughnuts to your liking.