Start the day before you want to fry the doughnuts. To make the tangzhong, place the flour in a small pan and gradually whisk in the milk, making sure that there are no lumps.
Place over a medium heat and cook, whisking constantly, until it has thickened to a wallpaper paste-like consistency. Remove from the heat, scrape into a bowl and cover with clingfilm directly on the surface, set aside to cool.
Once the tangzhong has cooled, place the plain and bread flours, sugar, salt and yeast in the bowl of a stand mixer and mix to combine.
Add the softened vegan butter, milk, cornflour and water mixture and all of the cooled tangzhong; mix until it forms a shaggy dough and there are no lumps of butter.
Mix on a high speed until the dough is very smooth and stretchy, about 5-10 minutes. It is quite a soft, sticky dough, resist the urge to add extra flour. You should be able to stretch a section of the dough thin enough to see light through it without it breaking (the windowpane test).
If you don't have a stand mixer then you can knead the dough by hand on an un-floured surface but it is very sticky and may take a while!
Place the kneaded dough in a lightly oiled bowl and cover. Place in the fridge overnight to rise. (You can leave it to rise at room temperature for 1-2 hours, until doubled in size instead but the flavour wont be as good and the chilled dough is easier to roll out.)
The following day, take the dough out of the fridge and punch it down. Knead very briefly to knock out any air bubbles then roll out on a floured surface to about 1cm thick. They will puff up a lot when you fry them so you don't want the dough to be too thick at this stage.
Use an approx 7cm/2.75 inch round pastry cutter to cut out as many circles of dough as you can, then use a 3.5cm/1.4 inch one to cut a hole out of the centre of each (I used the wide end of a large piping nozzle as I didn't have a cutter small enough. Something like a bottle cap may also work). Dip the cutters in flour first to stop them sticking.
Knead together the scraps and re-roll to cut out more doughnuts (you can either save the holes to fry or add them to the scraps to re-roll). Do not re-roll more than once however as you will end up with tough doughnuts. If you have lots of scraps after the second roll you can save them and use to make bread rolls if you like. You should end up with about 22 doughnuts and holes.
Spin each doughnut around your finger to widen the hole a little then place each one on it's own square of baking parchment well spaced apart on a baking sheet. For the doughnut holes you can place 3 or 4 on the same square of parchment provided they are spaced apart. Cover loosely with oiled clingfilm or baking parchment and set aside to rise for about 30-45 minutes until puffy.
Meanwhile, make the glaze. Melt the vegan butter in a pan over a low heat. Place the sifted icing sugar in a wide, microwave safe bowl and stir in the melted butter and vanilla extract. Gradually whisk in enough warm milk to form a thick but runny glaze. Set aside. If the glaze sets before you are ready to use it, simply heat it up for a few seconds in the microwave until it becomes runny again.
Heat a deep fat fryer or large pan of oil to 180C/350F. Use a fish slice/spatula to lift a doughnut on it's square of parchment into the hot oil then use a pair of tongs to lift the parchment out. This is the best way to transfer the doughnuts to the oil without them becoming misshapen or deflated.
Fry up to three doughnuts at a time for about a minute per side until golden. If you want to be certain that they are cooked through then you can use a probe thermometer to check - they should measure about 100C/212F inside.
Use a slotted spoon to lift the doughnuts out onto a plate lined with kitchen paper to drain.
If you are frying the doughnut holes too then they will take about 30 seconds per side. Only fry a couple at a time as they can be tricky to flip over and you may have to hold it with tongs to ensure that they cook on both sides.
Dip the doughnuts into the glaze while they are still warm then place them on a wire rack set over a roasting tin/baking tray to catch any drips. The doughnut holes can be dipped in the glaze too or can be rolled in cinnamon sugar instead.
Allow the glaze to set then eat straight away. They really must be eaten on the day they are made for the best experience, though if you do have any leftovers they can be stored in an airtight container and rewarmed in the microwave for 10 seconds to soften them up before eating the following day. By the third day the only thing they will be good for is bread pudding.