Sift the confectioners' sugar into a medium bowl. Slowly stir in 3 tablespoons of milk and the vanilla extract until the mixture is smooth. If the glaze isn't thin enough, stir in 1 additional tablespoon of milk. Cover the glaze with plastic wrap and set it aside while you make the doughnut holes.
Make the doughnut holes:
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.) 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, (I don't have a thermometer but I threw in a tiny piece of dough to see if it fried it sufficiently) use an ice cream scoop to drop about 1 tablespoon scoops of dough into the oil, careful not to overcrowd the pan. 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 pour the glaze on top of them and let cool so the sugar gets hard and delicious.
This is the recipe I would have used if I knew that yeast made a difference. For those of you who aren't experts on bread products, yeast allows the dough to become airy and light which is the number one quality of a donut.
2 cups all-purpose flour
1 1/2 cups sugar
2 teaspoons baking powder
1 teaspoon ground cinnamon
1/2 teaspoon ground nutmeg
1/2 teaspoon kosher salt
1 extra-large egg, lightly beaten
1 1/4 cups whole milk
2 tablespoons unsalted butter, melted
2 teaspoons pure vanilla extract
Into a large bowl, sift together the flour, sugar, baking powder, cinnamon, nutmeg, and salt. In a small bowl, whisk together the egg, milk, melted butter, and vanilla. Stir the wet mixture into the dry ingredients until just combined. Then fry the dough for two minutes on each side until cooked through. Let cool and then pour glaze over them.