Heat the milk until it is warm but not hot, about 100 degrees F In the bowl of a stand mixer fitted with the dough hook, combine milk with the yeast. Stir lightly, and let sit until the mixture is foamy, about 5 minutes.
Beat the eggs, butter, sugar and salt into the yeast mixture. Add half of the flour (2 cups plus 2 tablespoons), and mix until combined. Mix in the rest of the flour until the dough pulls away from the sides of the bowl. Add remaining flour; 2 cups plus 2 tablespoons. If the dough is too wet, add more flour 2 tablespoons at a time. The dough will probably become too thick to beat; when it is, transfer it to a floured surface, and gently knead it until smooth. Grease a large bowl with a little oil. Transfer the dough to the bowl, and cover. Let rise at room temperature, in a warm place, until it doubles in size, about 1 hour.
Lightly flour two rimless baking sheets. Turn the dough out onto a well-floured surface, and using a well floured rolling pin, roll dough into a 16’’ x 12’’ rectangle; it should be about 1/4-inch thick. Working quickly, and using a floured pizza wheel or pastry cutter, measure dough, then slice dough in half lengthwise. Measure dough, then slice dough into 2-inch sections crosswise. You will now have 12 long strips. Using a thin spatula to assist you, transfer cut dough to baking sheets, leaving about 4’’ of space between each one. Using your floured pastry cutter, slice a slit lengthwise through each piece of dough leaving 1/2’’ of uncut dough on both ends. Working with one piece of dough at a time, gently lift one end carefully separating the center. Twist the dough twice and place the end back down. Repeat with remaining dough.
Let twists rise in a warm place until they've puffed, about 35 minutes. About 15 minutes before they are done rising, Attach a deep-frying thermometer to the side of a large heavy-bottomed pot or Dutch oven; heat 2-inches of oil over medium-high, to 375 degrees F.
Prepare two cooling racks, topping them with paper towels and placing them over baking sheets. Using a metal spatula carefully add the doughnuts to the oil, a few at a time. When the bottoms are deep golden, about 1 to 2 minutes, use tongs to flip; cooking until they’re golden on both sides.Transfer the doughnuts to the prepared racks, repeating with the rest of the dough, ensuring the oil remains at 375F while frying.
In a small saucepan over low warm milk to a simmer. Pour warm milk into a medium heatproof bowl. Whisk in confectioners sugar, vanilla and salt until smooth and glossy. Place bowl over a pot of simmering water until thin and smooth, about 4 to 6 minutes. Remove from heat. Using your fingers, or tongs fully submerge each doughnut in the glaze. Place each glazed doughnut on a cooling rack placed over a baking sheet. Allow glaze to dry for at least 1 hour. Serve within a day.