In the bowl of your stand mixer, combine the warm milk and ½ teaspoon of the granulated sugar. Sprinkle the active dry yeast over the top. Let it sit for 5-10 minutes until foamy.

Add the remaining granulated sugar, egg, melted butter, and salt to the yeast mixture. Mix on low speed with the dough hook attachment until just combined.

With the mixer running on low, gradually add 2 ½ cups of the all-purpose flour, one cup at a time, allowing each addition to incorporate before adding more.

Increase the mixer speed to medium-low and knead for 5-7 minutes until the dough is smooth and elastic.

Lightly grease a large bowl. Transfer the dough and let it rise in a warm place for 1-1.5 hours, or until doubled in size.

Gently punch down the dough and turn it out onto a lightly floured surface. Roll the dough to about ½-inch thickness.

Use a donut cutter to cut out the donuts and holes. Place them onto individual squares of parchment paper.

Let the cut donuts rise again for 30-45 minutes until puffy.

Pour 6-8 cups of vegetable oil in a deep pot and heat to 350°F (175°C).

Carefully fry 2-3 donuts for 1-2 minutes per side until golden brown. Drain on a wire rack.

In a shallow bowl, whisk powdered sugar, milk, and vanilla until smooth. Glaze the warm donuts and let set.