Tip the flour, caster sugar, yeast and 1 tsp salt into a bowl, then mix to combine. Pour in 100ml water, the beaten egg and vanilla, and bring everything together into a rough dough using your hands, ensuring that all the flour is incorporated. Or, do this using a stand mixer. Cover with a clean, damp tea towel and set aside to rest for 30 mins.

Tip the dough onto a clean surface and knead for 10-15 mins by hand (or, knead for 5-8 mins in a stand mixer fitted with the dough hook on low speed) until elastic and glossy. Knead the softened butter into the dough using your hands, about 5 mins. At first, it will look like it’s not coming together, but keep working the dough through your fingers and it will become smooth and glossy. Form the dough into a ball and return it to the bowl. Cover and leave in a warm place to rise for 2-3 hrs, or until nearly doubled in size. Line a baking tray with baking parchment.

Roll the dough out on a lightly floured surface to a 2cm thickness, then use a 9cm cutter to stamp out as many circles as you can. Use a 4cm cutter to stamp out the middles. Re-roll the trimmings (including the middles) and stamp out more (you should end up with about eight ring doughnuts). Re-roll any remaining trimmings and continue cutting until all the dough is used. Put the doughnuts on the prepared tray, cover with a clean tea towel and leave to rise in a warm place until doubled in size for another 2-3 hrs, or chill overnight. The lightness of the doughnuts depends on this second rise – a longer rise in the fridge will result in more flavour.

Fill a deep-fat fryer with sunflower oil following the manufacturer’s instructions, or fill a heavy-based pan no more than a third full. Heat to 170C, or until a cube of bread dropped in browns within 30 seconds. Carefully lift two or three doughnuts from the tray and slide them into the hot oil using a floured spatula. Fry for 2 mins on each side until puffed up and golden brown, then remove to a sheet of kitchen paper using a slotted spoon or tongs. Leave to cool. Repeat with the remaining doughnuts.

For the glaze, sift the icing sugar into a bowl and mix with 2 tbsp water until you have a runny icing – you might need to add another teaspoon of water if your glaze is a little stiff. Dip the smoother side of the doughnuts into the glaze and serve straightaway. Will keep in an airtight container for a up to a day.