In a large mixing bowl, or the bowl of your stand mixer, add the strong white bread flour, yeast, sugar and salt. When adding the yeast to the bowl, do not pour it directly on top of the salt (or do not pour the salt directly on top of the yeast) as this can kill the yeast. Then stir them all together
In a pan on a low heat, warm the milk and butter together until the butter has melted, set aside to cool for 10 minutes. Then add the vanilla extract and egg to the milk and butter mixture and whisk it in
With your stand mixer running on a slow speed, or use a wooden spoon to stir if mixing by hand, pour the milk mixture into the bowl of flour to form a dough
Knead the dough for 8 minutes, either in your stand mixer with the dough hook attachment, or knead by hand on a lightly floured surface. The dough will be quite sticky so do not be tempted to add lots of flour to it
Place the kneaded dough into a lightly oiled bowl, cover with cling film or a tea towel and leave to rise for 2 hours or until doubled in size
Tip the dough out of the bowl and roll it around on a floured work surface with your hands to knock the air out of it
Divide the dough into portions, you can weigh them if you like for accuracy, and roll each portion into a ball using your hands. The dough will still be slightly sticky so flour your work surface when rolling them
Place the balls of dough onto well floured and lined baking trays well spaced apart, cover them with oiled cling film and leave to rise for an hour or until doubled in size
Prepare you pan of hot oil, or your deep fat fryer, to a temperature of 170-180C (350-375F)
Handle the balls of dough very gently, I find using a large slotted spoon or spider strainer to lower them carefully into the hot oil the best method. Fry the donuts in batches of 3 or 4 on either side until golden (this should take 1 ½ - 2 minutes on each side and you can tell that each side is done as the oil will bubble much less around the donut), use the slotted spoon/strainer to flip them over. If they are browning too fast or going too dark, turn the temperature of your oil down by 5-10 degrees
Set them on some kitchen towel until you can safely handle them without burning your hands, then toss them in the caster sugar in a bowl while they are still warm
Make a small hole in the side of the donuts (use a knife, chopstick, or a piping nozzle) and pipe the jam into them until they feel heavier, you feel resistance and the jam starts to ooze out slightly
They are best eaten straight away, but can be stored for up to 2 days