- For the doughnuts, place all of the ingredients in the bowl of a mixer fitted with the hook attachment and mix on low speed until blended, then increase speed one level and knead until the dough is smooth and elastic, about 4 minutes. Alternatively, you can blend the ingredients by hand, and then turn out onto a floured surface to knead until smooth, about 6 minutes. Place the dough into a bowl, cover with plastic wrap and leave to rise for an hour or until doubled.
- To cut the doughnuts, turn the risen dough out onto a floured surface and roll to 1/2-inch thickness. Cut doughnuts using a 3-inch cutter and cut a hole out of the centre of each. Re-roll any scraps (the holes can be re-worked into the dough, or fried as doughnut holes). Cover the doughnuts with a tea towel and let rise for 30 minutes.
- While the doughnuts are rising, prepare the glaze. Whisk the melted butter, with the icing sugar, vanilla and 3 tbsp of the milk until smooth, adding more milk if needed to make a thin glaze (it should be quite fluid).
- Preheat vegetable oil to 350°F in a tabletop fryer, or in a deep pot over medium high heat (fill the pot with 2 inches of oil). Carefully drop in a few doughnuts, leaving enough space so that they do not touch, and cook for 4 to 5 minutes, turning halfway through cooking. Lift the doughnuts onto a cooling rack with a paper-towel lined tray underneath to drain and cool for 5 minutes, and continue to cook the remaining doughnuts in batches.
- After each batch of doughnuts has cooled just a little (they can still be a touch warm), dip them into the glaze so that they are fully covered. Shake off excess glaze and place of a cooling rack with a lined tray underneath to allow the glaze to set for about 15 minutes.
The doughnuts are ideally enjoyed within an hour of cooking, but can still be appreciated the day they are made.