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Combine the milk, melted butter, yeast, 1 tsp of the sugar and the vanilla extract in the bowl of a stand mixer. Leave for 8-10 mins for the yeast to activate.
In a separate bowl, combine the remaining sugar, flour, ½ tsp salt and the cinnamon, if using. Mix the beaten egg into the milk mixture, then fold in the dry ingredients. Knead with a dough hook on a medium speed for 5-8 mins until the dough is smooth and elastic. You can also knead by hand for 10-12 mins, until smooth. It’s a very sticky dough, so be patient with it.
Transfer to a lightly oiled bowl, cover with a clean tea towel and leave in a warm place for 1 hr 30 mins until the dough has doubled in size. Tip the dough out onto a lightly floured surface and roll out to around 1.5cm thick. Cut out your doughnuts using a doughnut cutter or two cutters (1 x 7.5cm and 1 x 2.5cm for the middle). Put the doughnuts on a lined baking sheet, along with the doughnut centres, if you like, and cover with a clean tea towel. Leave to rise for 40 mins or overnight in the fridge (the doughnuts will hold their shape better if proved in the fridge).
When ready to cook, place 2-3 doughnuts (and their centres, if keeping) in the air-fryer basket and cook at 180C for 5-6 mins, following manufacturer's instructions, until golden. (Bear in mind that some air fryers generate more intense heat, so keep checking to make sure the doughnuts don't burn.) Remove from the basket and leave to cool on a wire rack while you cook the remaining doughnuts. You can place a sheet of baking parchment at the bottom of your basket if you’re worried about the doughnuts sticking and to prevent indents from the basket forming on the doughnuts.
While the final batch is cooking, sift the icing sugar into a bowl and stir in the milk and vanilla extract. When the doughnuts have cooled, dip the top of each one in the glaze. Leave on the wire rack for the glaze to set. Best eaten on the day but will keep for up to 24 hours in an airtight container.