Place all the ingredients for the dough, including the butter, into a stand mixer and using a dough hook mix for approximately 10 - 15 minutes until the dough forms a ball and comes away from the sides of the bowl. Continue to beat on a higher setting until the dough is glossy, smooth and elastic. Cover the bowl with a tea towel and leave it to rise until it has doubled in size (approximately 1 hour).
Once the dough has risen, tip it from the bowl and knock it back, kneading it a few times to get the air out. Place it back in the bowl, cover again and allow to prove a second time, until it has again doubled in size.
Make the custard: Place the milk into a pan. Cut open the vanilla pod and scrape out the seeds into the milk, and add the pod back in. Place the egg yolks, sugar and flour into a bowl and whisk together to combine Bring the milk to scalding point and then remove from the heat. Slowly pour the hot milk into the egg mixture, whisking continuously. Once all added, clean out the pan and then pour the mixture through a fine sieve back in and place it back on the heat. Gently warm and continue to whisk until the custard thickens.
Once thickened, remove into a bowl to cool.
Once the dough has doubled in size. Then cut the dough into roughly 50g pieces and roll them out into neat balls. Place them onto a well floured tray and spaced apart, and dust again with lots of flour and cover the whole tray loosely with cling film. Place the trays in a warm place to let the doughnuts have their final prove, until they are puffed up and risen.
Using an electric whisk beat together the double cream until it reaches medium peaks. Then spoon this into the cooled custard and fold together. Spoon the custard into a piping bag with a small round nozzle attached. Seal the piping bag and keep in the fridge until later.
Once the doughnuts have had their final prove they are then ready to fry. Place the oil into a high sided saucepan and heat until it reaches 180 C. Fill the base of a baking tray with caster sugar for dusting.
Using floured hands, carefully lift the doughnuts off the tray being careful not to knock the air out of them. Drop into the oil and allow the doughnut to cook on one side for approximately 2 minutes, then flip in the oil and cook the other side. Do these in batches. Once cooked remove from the oil and momentarily drain on some paper, before adding to the tray of caster sugar and dusting liberally.
Once all the doughnuts have been fried and coated in sugar, fill with custard. Using a chopstick, make a hole in the side of each doughnut to the centre. Take the piping bag and insert the nozzle into the hole and fill the centre, and finish with an extra bit of custard on the outside before serving.