Whisk together the yeast, 1/2 tsp of sugar and the warm water. Set aside for 5 minutes until the yeast has dissolved and the mixture is bubbly.
Meanwhile, whisk together the flour and salt in the bottom of your stand mixer bowl.
Whisk the egg yolk, milk and vanilla into the yeast mixture, and pour it into the flour mixture. Bring together into a rough dough with a fork.
Fit the dough hook and knead the dough for 5 minutes, gradually adding the butter and scraping down the sizes if needed. You should be left with quite wet, sticky dough.
Cover the stand mixer bowl with food wrap and leave in a warm place for 2 hours until the dough has at least doubled.
Scrape the dough out onto a very well floured surface and knock out any air bubbles. Using a well floured rolling pin, roll the dough out to a 1/4 inch thickness.
Use your largest and smallest cookie cutters to punch out doughnut shapes: you can bring the dough together and roll it out again to get more doughnuts out of the scraps.
Transfer the doughnuts to a well floured tray and cover with food wrap. Leave to rest for 30 minutes.
Meanwhile, heat at least 4 inches of oil in the bottom of a large, heavy bottomed saucepan or casserole to 360F, and in a shallow dish mix together the sugar and cinnamon for the coating.
Working in batches and using a slotted spoon to transfer the dough to the hot oil, fry the doughnuts for 2-3 minutes on each side until deeply golden. Once puffed and cooked, transfer them first to a kitchen paper lined plate, then roll them in the cinnamon sugar coating about 30 seconds later. If you transfer them straight to the sugar without removing any excess oil you’ll get an uneven coating, but if you allow too much of the excess oil to soak into the doughnuts, the sugar won’t stick properly.