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 tablespoons of the milk until smooth, adding more milk if needed to make a thin glaze (it should be quite fluid).
Preheat vegetable oil to 350F 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.