American
Mix 125 ml of the milk with the sugar and yeast, then add the flour, egg and salt and knead for 5 minutes.
Knead in the soft butter a little at a time for 5 more minutes until the dough is smooth, then cover and let it rise for 60 minutes until doubled.
Roll the dough 1.5 cm thick, cut 8 rings with an 8 cm cutter and a 3 cm cutter for the holes, and let them rise on floured baking paper for 30 minutes.
Heat the oil in a deep pot to 175C.
Fry the donuts and holes a few at a time for about 1 minute per side until golden, then drain on a rack.
Whisk the powdered sugar with the remaining 45 ml milk and the vanilla until smooth.
Dip the warm donuts into the glaze, turn to coat, and let the glaze set on a rack for 10 minutes.