Pour the milk in a pan and warm gently until it is about body temperature. It should feel neither warm nor cold when you dip your finger in.
To mix by hand, rub the yeast into the flour using your fingertips as if making a crumble. Rub in the butter, then add the sugar and salt, then the eggs and milk.
Let the dough rest for 1 hour.
Turn the dough onto a lightly floured counter and form it into 30 tight balls, weighing about 1 ounce each.
Lightly oil a baking tray and lightly dust it with flour. Arrange the donuts on top (seam-side down), leaving space between them so that they will not touch as they rise. Cover tray with lint-free dishtowel and let rise until they have doubled in size, about 45 minutes.
Pour oil into an 8-inch saucepan and place over medium eat. When the oil is hot, allow about 15 minutes to reach 350F, use a slotted spoon to lift the first donut and flip it into the oil. It should start sizzling right away.
Add more until you fill the pan (I fry a maximum of 5 at a time). Fry for 30-45 seconds, until the donuts start to color. Then turn over and fry for the same amount of time.
Use a slotted spoon to remove from oil and drain on paper towels. Let cool.
For plain donuts, simple roll them in superfine sugar. or filled donuts, spoon the filling of your choice into a pastry bad with a small nozzle and fill by inserting the tip into the donut ad one side.