Heat the milk in a small saucepan until just simmering. Alternatively, microwave until hot.
Add the hot milk to a large mixing bowl. Add the butter and sugar. Once the butter has melted, whisk to combine.
The milk mixture should be warm, but not hot, to the touch. Whisk in the active starter.
Add the egg and whisk to combine.
Add the flour and salt to the bowl and use a wooden spoon or spatula to mix until a shaggy dough forms.
Cover and set the dough aside. Every twenty minutes over the course of the next hour, do a round of stretches and folds. After this time, the dough should be soft and form a ball easily.
Cover and set the dough in a warm or room-temperature place to rise until doubled in size. The time needed will depend on ambient temperature and the strength of your starter, but it should need anywhere from 4-8 hours.
Once the dough has risen fully, chill for at least 2 hours. This is optional but will make it much easier to shape.
Shape the chilled dough. Turn it out onto a lightly floured surface, flatten slightly, then roll into a spiral log shape. Turn the log so that the short end faces you, seam-side up, and roll it into a spiral again.
Flip the dough seam-side down and tuck the edges under it. Rotate the dough with your hands to make a ball with some tension on the outside of the dough, the same as you would for a sourdough boule.
Roll the dough out to 1.5cm (⅔ in.). Use a round biscuit cutter (7cm / 2 ¾ in. diameter) to cut rounds in the dough. Use the large end of a piping tip to cut holes in the centre of each round.
Carefully lift the cut donuts and place them onto a parchment paper lined baking sheet. Repeat rolling and cutting the dough until it's all been used.
Cover the donuts with a damp tea towel and set aside to rise again until puffy looking, 1-2 hours.
Once the donuts have risen, heat the oil for deep frying. It should be 375°F (190°C). Line a large plate with paper towels and prepare the cinnamon sugar.
Fry the doughnuts for just over 1 minute each side. Flip when you see the bottom half has browned, but if the oil is the correct temperature, it should take 2 minutes per donut. Carefully remove from the oil with a slotted spoon and set on the paper-lined plate to dry.
While the donuts are still warm, coat them in cinnamon sugar. Repeat with step 16 until all of the donuts are fried.
For the donut holes, fry for just under 1 minute, stirring constantly to ensure even browning. Dry on the plate before rolling in cinnamon sugar.
Sourdough donuts are best eaten fresh, but will keep for a day or two in a sealed container at room temperature.
Notes
If your eggs or any of the other ingredients are cold, it will cause the dough to rise very slowly. Make sure everything is at room temperature to avoid any delays.