Prepare Oil and Workspace: Heat oil in a deep fryer or large pot to 360℉. place a wire cooling rack inside a baking sheet and place paper towels on top to absorb excess grease after frying.
Mix Dry Ingredients: In a large bowl or mixer, whisk together the flour, sugar, baking powder, baking soda, cinnamon (if using), and salt.
Add Wet Ingredients: Add the sourdough discard, oil, egg, and milk to the dry ingredients. Using a dough hook, mix until just combined to form a wet dough. Be careful not to over-flour, as this can make the donuts too thick.
Knead the Dough: Turn the dough onto a floured surface and knead for 3-4 minutes. Roll out to a thickness of 2 inches.
Cut Out Donuts: Use a large biscuit cutter to cut out donuts. For the center holes, you can use a cap from a bottle or any other small round cutter. These will make perfect donut holes to fry.
Fry the Donuts: Fry the donuts in the heated oil until golden brown, about 1-2 minutes per side. Place the fried donuts on the prepared cooling rack to drain excess oil.
Glaze the Donuts: Prepare a glaze with confectioners' sugar, milk, and vanilla extract. Dip the warm donuts in the glaze and place back on the rack to set for about 10 minutes. This allows the glaze to adhere to the donuts.