In the bowl of a stand mixer, combine 2 cups of the flour, the yeast, sugar and salt and stir well.
Add the melted butter and stir until crumbly, then stir in the egg. Add the warm milk and stir to make a very soft dough, then add remaining flour about 2 tablespoons at a time until dough just barely sticks to the sides of the bowl when stirred.
Using the dough hook, knead the dough until smooth and elastic. Dough should leave the sides of the bowl as it is kneaded. If it doesn't, add a little more flour.
Cover the bowl and allow dough to rise for about 45 minutes or until doubled in bulk.
Turn the risen dough out onto a well-floured surface, and roll it to 1/2-inch thickness. Cut out the doughnuts with a 3 inch round cutter doughnut cutter and use a smaller cutter to cut out the hole the doughnut holes.
Put the doughnuts on a floured baking sheet and let rise for about 25 minutes or until they are slightly puffed up and delicate. Keep them very loosely covered with a piece of greased plastic wrap.
While the dough is rising, I recommend getting set up for the glaze. Put a wide saucepan on the stove and add 1/4 cup of milk. Have ready 1 teaspoon of vanilla and 2 cups of powdered sugar.
About 15 minutes before the doughnuts are done rising, put the oil in a heavy-bottomed pot or Dutch oven over medium heat, and heat it to 375. If using cast iron, you may want to heat only to 350. Meanwhile, line cooling racks, baking sheets or plates with paper towels.
Carefully add the doughnuts to the oil, a few at a time. Adjusting heat if necessary. After about 45 seconds, check that the doughnuts have browned and then flip to the other side. Fry until other side is brown, then drain on paper towels.
Cook the doughnut holes the same way, keeping a close eye on them and allowing them to brown all over..
While doughnuts are frying and cooling, heat milk in a large saucepan just until it simmers, then stir in vanilla and powdered sugar until smooth.
Dip the fried doughnuts in the glaze, covering them on both sides, then set on a wire rack set over some parchment or foil.