Melt the butter in a small sauce pan on a low heat then whisk in milk and continue warming until steaming.
Add the warm water to the hot milk then transfer mix to the bowl of a stand mixer.
Sprinkle the yeast in the bowl allowing it to sit and bubble for 5 minutes.
Add in the egg, sugar, salt and about half of the flour.
Using the dough hook attachment, mix the ingredients together on low speed until just combined then add in enough of the remaining flour to give you a formed, but slightly sticky dough. Leaving the dough sticky will ensure that your doughnuts are airy and soft!
Knead the dough for about 7 minutes.
Grease a large bowl and place the dough inside. Cover with cling wrap, place in a warm spot and allow to rise for 1 hour, or until it has doubled in size
Roll the dough out on a lightly floured surface, until it is about 1cm thick.
Use an 5cm cutter to cut out the donuts.
Place the donuts on a floured baking tray, and cover with cling wrap. Place in a warm place for another 15 minutes.
Meanwhile, heat your oil to 180 degrees in a heavy base saucepan. Sacrifice a donut and test the oil before frying a whole batch.
Fry the doughnuts for about 2 minutes on each side and place on a roasting rack to cool before glazing.
To make the glaze, whisk together the milk and peanut butter until smooth.
Whisk in the icing sugar then set aside.
Fill a piping bag with a 5mm round nozzle with the jam.
Take a chopstick and poke a hole through the side of the donuts, all the way through to the other side.
Fill the hole with the jam, using both holes to ensure an even distribution.
Cover the donuts with the glaze.