Put the flour, caster sugar, salt, yeast, eggs and water into the bowl of a stand mixer fitted with the dough hook. Mix on medium speed for around 8 minutes, until the dough comes away from the sides of the bowl and wraps around the dough hook.
Turn off the mixer and let the dough rest for 1 minute. Slowly add the butter in small lumps with the mixer on medium, don't add the next batch of butter until the previous amount has been completely mixed in.
Once all the butter is added, mix on high for 5 minutes, until the dough is glossy, smooth and very elastic when pulled. Cover the bowl with clingfilm and leave to prove until it has doubled in size. Knock back the dough, then re-cover the bowl and put into the fridge to chill overnight.
The next day, take the dough out of the fridge and roll it out to a thickness of 2cm. Take a cookie cutter that is around 8-9cm in diameter and another that is 2cm, stamp out large circles to make the doughnut and then using the smaller cutter, stamp out a smaller circle to create a ring doughnut.
Roll up the scraps of dough and roll them back out and stamp out more doughnuts, keep doing this until you have used nearly all the dough. This should make 9 doughnuts.
Place the doughnuts onto a baking tray lined with greaseproof paper, spaced widely apart, to allow for them to prove without sticking together. The doughnuts need to double in size (I made these during our heatwave and this only took just over an hour and a half, but in cooler weather this can take anything up to 4 hours as the dough will be cool from the overnight refrigeration).
Get a heavy-based saucepan and fill it up to the halfway point with the oil and heat the oil to 180° C. When the oil is ready, gently pick up the doughnuts, trying not to drag them or to distort the shape too much. You will probably want to fry just one doughnut first, to check the oil and to get an idea of timings, about 2 minutes on each side should be enough. Remove the doughnut from the oil with a slotted spoon and place on a cake rack lined with kitchen towel.
After the first doughnut, you should then be able to fry two doughnuts at a time, but make sure the oil is always back to 180C before frying. If the oil is too hot, the doughnut will be raw in middle and burnt on the outside and if it is too cold, the doughnuts will be greasy. Continue until all of the doughnuts have been fried.
Allow the doughnuts to cool while you make the icing, combine the sifted icing sugar with the maple syrup in a bowl that is wide enough to fit a doughnut in. Once evenly mixed, you will have a very thick icing, dip the top of the doughnut in the icing until it is evenly coated. Then set back on a cake rack and sprinkle generously with the maple flakes.
The finished doughnuts are best enjoyed within 24 hours or they will start to become greasy and they won’t taste as good - but they can still be eaten.