In a small saucepan, add the milk and butter/margarine. Warm over a low heat until the butter/margarine starts to melt. Remove from the heat and allow to cool a little - you don't want it to be too hot before adding to the other ingredients.
In a large bowl, sift in the flour and sugar. Add the yeast and gently mix to combine. Make a well in the middle of the dry ingredients, then add the milk mixture and egg. Mix together to create a soft, slightly sticky dough - you may need to use your hands!
Once the dough has come together, tip it onto a well-floured surface and begin to knead with floured hands, adding more flour as needed. Knead for around 10 minutes, until you have a smooth, stretchy dough.
Place the dough into a oiled bowl (to prevent it from sticking) and cover with an oiled piece of cling film, or a damp tea towel. Leave to rise in a draught-free location for about an hour, or until it has doubled in size.
Once the dough has risen, knock it back and briefly knead it on a lightly floured surface. Divide your dough into 12 even sized balls, rolling them between your hands to shape them. Place your dough balls onto individual squares of greaseproof paper (approx. 10cm² in size) with a bit of space between them - preferably on a tray or board - and cover again with a piece of oiled cling film or a damp tea towel. Leave in a draught-free place for about an hour, until they have puffed up to about double their size.
In a large pan, heat your oil to 180°c/350°F (use a thermometer for accuracy). You want your oil to be deep enough for your doughnuts to comfortably fry, without sticking to the bottom. Depending on your pan size, you may need to use more/less oil.
Using the greaseproof paper square, lift your doughnut to the oil, and gently drop it in, exercising extreme caution. Allow the donut to fry for about 90 seconds before flipping it over, allowing the underside to fry for about 90 seconds. Depending on the size of your pan, you should be able to fry the donuts in small batches; just be careful not to overcrowd your pan, or you risk the doughnuts sticking together and the oil dropping in temperature too quickly.
Once your doughnuts are golden brown on both sides, use a slotted spoon to remove them, holding them above the oil for about 10 seconds (to allow the oil to drip sufficiently) before transferring to a wire rack to cool. Continue until all your doughnuts are cooked, whilst continuing to monitor your oil temperature and adjusting the heat as needed.
Spoon your jam into a piping bag with a large circular nozzle. Once your doughnuts are cool enough to touch, poke a little hole with the end of a wooden spoon to the side of each doughnut, before piping your jam into the hole. It can be a bit of trial and error (and personal preference) getting the right amount of jam in each donut, but around a teaspoonful amount is usually ideal.
Roll the doughnuts in some extra caster sugar to finish.
Notes
Freshly made donuts are best enjoyed the same day they are made.