To make the dough: In a saucepan add milk and bring to a scald. Add butter and melt, add vanilla paste and temper in egg. Set aside. In the bowl of a mixer fitted with a dough hook combine all dry ingredients including yeast. Add milk mixture and set mixer on medium. Mix until a slightly tacky dough is formed, add additional flour in 1/4 cup increments if still too wet.
Turn dough out onto a floured surface and lightly knead a few times. Cover and let rest at room temp for 30 mins. After 30 mins check rise, if the dough has risen at least by half cut in half and roll out to about 1 cm thick. Cover again and let rise for 20 mins. After second rise cut out doughnuts (I use round cookie cutters of different sizes but I have used cups in the past, there's also doughnut cutters.)
Set aside on a floured surface and cover again to let rise. Repeat this with the other half, any extra pieces I roll into doughnut holes or cut into squares.
To fry: Heat the oil in a heavy frying pan or skillet to 350°F. Carefully place the doughnuts in the oil, 2 or 3 at a time, and fry until golden brown. Turn over and cook the second side; each side should take no more than a minute. Remove from the oil with a slotted spoon and drain on a wire rack.
Fill or frost doughnuts as desired, using your choice of sugar topping or glaze.