In a small bowl mix the yeast with the milk and sugar. Let sit until foamy, about 10 minutes. In a mixing bowl of a stand mixer fitted with the dough hook, combine flour and salt. When yeast is ready, add that to the mixing bowl.
Add the egg and butter. Begin to mix the ingredients into a soft, but not sticky, dough. (Add a little extra milk or extra flour, 1 tablespoon at a time as needed). Knead dough until smooth and elastic. Transfer the dough to a lightly oiled bowl, cover, and let rise until doubled in size, about 2 hours.
Punch down the dough. Turn out onto a lightly floured surface and knead for a few seconds. With a lightly floured rolling pin, gradually roll out the dough to about 1/2-inch thick. (When rolling dough, let it rest periodically to relax). Cut out into 3" rounds with a lightly floured biscuit cutter. Re-roll the scraps to make more rounds.
Place the doughnuts on lightly floured baking sheets, spacing them apart, and cover lightly with a dry towel. Let rise in a warm spot until doubled in size, about 20 minutes.
Using a deep fryer (or fill a heavy pot with 3" vegetable oil) and heat to 325°F Fry the doughnuts, a few at a time, until golden and puffed, 4-5 minutes, turning frequently using a slotted spoon.
Lift the doughnuts from the oil and place onto a cooling rack. Once cooled, fill a pastry bag using your filling of choice. Insert the tip into the end of each doughnut and pipe approximately 1-2 tsp filling into them. Dust with confectioners' sugar, coat with chocolate icing, or dip in glaze. Best eaten the same day.