In a small bowl, sprinkle the yeast over the lukewarm milk. Stir briefly and allow to sit at room temperature until frothy, about 10 minutes.
In the bowl of a stand mixer fitted with a dough hook or a large bowl, combine the flour, sugar, and salt.
Mix in the frothy yeast milk mixture, egg, and butter until a soft dough forms.
Knead on a lightly floured surface just until elastic and smooth. Place in a large bowl, cover, and allow to rest at room temperature until doubled, 1-2 hours.
On a very lightly floured surface, divide the dough into 12 equal pieces.
Form each piece into a smooth ball, cover with a towel, and rest again at room temperature until puffed, 30 minutes to 1 hour.
After a second rest, roll one of the dough pieces into a thin rope about 12 inches (30.5 centimeters) long. Only use flour as needed to keep the dough from sticking.
Place the rope horizontally across the work surface in front of you.
Place your left hand on the left half of the rope and the right hand on the right half of the rope. Gently roll the left half of the rope upwards and the bottom half of the rope downwards in opposite directions until the rope twists 3-6 times.
Hold up each end of the rope so the center hangs in the air. It should start to naturally twist. Guide the rope to twist 3-6 times, then press together the ends to keep them from unraveling. Repeat with the remaining pieces of dough.
Cover the twisted dough pieces with a towel and allow to rest at room temperature for 30 minutes, until puffed.
In a wide bowl, stir together the 1/2 cup (100 grams) sugar and cinnamon until evenly combined.
Pour 2 inches (5 centimeters) of vegetable oil in a large saucepan and heat to 340˚F (170˚C), no higher.
Once heated, gently add two or three of the twisted doughnuts, taking care not to overcrowd.
Fry until lightly golden on the bottom, about 2 minutes, then flip to fry the other side until golden.
Transfer to a towel-lined platter, then immediately coat in the cinnamon sugar mixture.
Repeat with remaining doughnuts, keeping the temperature no higher than 340˚F (170˚C).
These doughnuts are best warm and served within a day.