Add the vegetable oil to a large, heavy-bottomed pot. (There should be at least 2 inches of oil in the pot and at least 2 inches between the top of the oil and the top of the pot.) Attach the deep-fry thermometer to the pot and begin heating the oil over medium heat to 350ºF. Line a baking sheet with paper towels.
In a small bowl, whisk together the milk and the egg.
In a separate medium bowl, whisk together the flour, sugar, baking powder and salt. Stir the milk-egg mixture into the dry ingredients, then stir in the melted butter, mixing until a soft dough forms.
Once the oil has reached 350ºF, use a small ice cream scoop to drop about 1 tablespoon scoops of dough into the oil, careful not to overcrowd the pan. (See Kelly's Notes.) Fry the doughnut holes, flipping them in the oil, for about 2 minutes or until they're golden brown. Using a slotted spoon, transfer the doughnut holes to the paper towel-lined baking sheet.
Allow the doughnut holes to cool slightly. Place a cooling rack atop a baking sheet, then one by one, dip the doughnut holes into the glaze and transfer them to the rack to allow the excess glaze to drip off. Serve immediately.