To make the dough, mix the bread flour, sugar, yeast and salt together in a bowl.
Add the milk, melted butter and vanilla extract and mix to form a soft, sticky dough.
Knead the dough on an unfloured surface for about 10 minutes, until it is smooth, supple and stretchy. It should still be a bit sticky but not wet. If it is too wet you can add a teeny bit more flour but be careful not to add too much or the doughnuts will be dry.
Place the dough in an oiled bowl, cover, and leave to rise until doubled in size. About 1 ½ to 2 hours in a warm spot or overnight in the fridge.
Punch down the risen dough and give it a brief knead to get rid of any air bubbles.
Divide the dough into 8 even pieces. Use a scale for accuracy.
On a lightly floured surface, shape each piece into a ball. To do this, gather the edges up and pinch them together. Keep going until you have a neat, well-sealed ball. Flip the ball over, cup it with your hand over the top and roll it until it is smooth.
Flatten the ball with your hand into a disc about 1 ½ - 2cm thick. Repeat with the rest of the pieces of dough. (If you want really fat doughnuts then you don't need to flatten them).
Place the discs of dough on a sheet of baking parchment on a baking sheet. Cover loosely and leave to rise until puffy, about 45 minutes. They are ready when the dough springs back slowly most of the way when you press it gently with a finger. If it springs back quickly then it needs a little longer, and if it deflates or doesn't spring back at all then the doughnuts are over-prooved and you will need to knock them back, reshape them and let them rise again.
When the doughnuts are almost ready, heat a deep fat fryer to 170°C/340°F. Fill a shallow bowl with caster sugar and line a plate with kitchen roll.
Cook the doughnuts two at a time. Slide them carefully into the hot oil; they are delicate so handle them as little as possible. Cook for 2 ½ minutes (use a timer), then flip them over and cook for another 2 ½ minutes. They should be deep golden.
Remove using a slotted spoon and drain on kitchen paper. Roll the doughnuts in the caster sugar while they are still warm.
Use a skewer or small sharp knife to make a hole in the side of each doughnut. Wiggle it around a little bit to make the hole inside a bit bigger.
Place the jam in a piping bag fitted with a medium round nozzle. A syringe also works well. Insert the nozzle into the hole you made in the side of the doughnut and pipe in a good splodge of jam. Serve as soon as possible.