About 15 minutes before the doughnuts are finished 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.