In a small pan, warm the milk until it's lukewarm. Set aside.
Sift the flour into a large bowl, add the yeast and sugar, stir to combine. Make a well in the middle of the dry ingredients and add the butter, milk and egg. Mix to form a soft dough.
On a floured surface, knead the dough for approximately 10 minutes, before transferring to a a lightly oil bowl and cover with a lightly oiled piece of clingfilm. Leave in a draft-free place for 1-2 hours or until the dough has doubled in size.
Once the dough has risen, knock it back and knead it lightly on a floured surface. For ring donuts, roll the dough to about 1cm thickness, before cutting to your desired size, either with a donut cutter or two round cutters (one smaller and one larger). Alternatively for filled donuts, roll the dough into equal sized balls, taking into consideration that the dough will rise further when choosing your desired donut size.
Transfer your donuts to an oiled tray/board/plates and cover with a piece of lightly oiled clingfilm. Leave in a draft-free place for about 1 hour until the donuts have doubled in size.
Fill a pan with around 1 litre of oil (sunflower or vegetable is ideal) and heat to 180°c. Once the oil is hot, carefully drop your risen donuts into the oil. The number of donuts you can fry at once will depend on your pan size and donut sizes. Just be cautious not to overcrowd your pan, as each donut added will lower the oil temperature and affect the overall frying.
As soon as your donut is golden on one side, flip it over with some chopsticks to fry the other side. Once both sides are golden, remove from the oil and place on some kitchen roll to absorb the excess oil.
If you plan to sugar your donuts, roll them in some caster sugar whilst still warm. Alternatively you can glazed them, which again should be done whilst still warm. Or if icing your donuts, wait until they are cool before finishing. For filled donuts, wait until they are cool enough to handle and make a small hole in the donut before using a piping bag to fill with jam, custard or Nutella.