In a large mixing bowl (or stand mixer bowl), combine the warm milk, active dry yeast, and 1 teaspoon of the granulated sugar. Stir gently and let sit for 5-10 minutes, until foamy.

Add the remaining granulated sugar, egg, melted and cooled unsalted butter, vanilla extract, and salt to the yeast mixture. Whisk until well combined.

Gradually add the all-purpose flour, 1 cup at a time, mixing until a shaggy dough forms. If using a stand mixer, attach the dough hook and knead on low speed for 5-7 minutes, until the dough is smooth and elastic. If kneading by hand, turn the dough out onto a lightly floured surface and knead for 8-10 minutes.

Lightly grease a clean bowl with a small amount of vegetable oil. Place the dough in the bowl, turning once to coat. Cover the bowl with plastic wrap or a clean kitchen towel and let rise in a warm place for 1-1.5 hours, or until doubled in size.

Once the dough has doubled, gently punch it down to release the air. Turn it out onto a lightly floured surface. Roll the dough to about 1/2-inch thickness. Using a donut cutter (or a 3-inch round cutter for the donut and a 1-inch cutter for the hole), cut out the donuts. Re-roll scraps once to cut out additional donuts.

Carefully transfer the cut donuts and donut holes to baking sheets lined with parchment paper, leaving some space between each. Cover loosely with plastic wrap or a kitchen towel and let rise for another 30-45 minutes, or until visibly puffy.

While the donuts are rising for the second time, pour vegetable oil into a large heavy-bottomed pot or Dutch oven, ensuring it is at least 3 inches deep. Heat the oil over medium heat to 350°F (175°C). Use a deep-fry thermometer to monitor the temperature.

Carefully slide 2-3 donuts (do not overcrowd the pot) into the hot oil using a slotted spoon or spatula. Fry for 1-2 minutes per side, until golden brown. Adjust heat as needed to maintain 350°F. Donut holes cook faster, about 30-60 seconds per side.

Using a slotted spoon or spider, remove the fried donuts and place them on a wire rack set over a baking sheet to drain excess oil. Repeat with remaining donuts and donut holes.

While the donuts are still warm, prepare the glaze. In a medium bowl, whisk together the sifted powdered sugar, milk, and vanilla extract until smooth.

Dip each warm donut into the glaze, turning to coat both sides. Place them back on the wire rack for the glaze to set. Serve immediately or once glaze is fully set.