In the bowl of a stand mixer, dissolve the yeast in warm water with a pinch of the granulated sugar and let it sit until frothy, about 5 to 10 minutes.

Add the warm milk, melted butter, beaten egg, remaining granulated sugar, and salt to the yeast mixture, stirring gently to combine.

Gradually add the flour to the wet ingredients, mixing on low speed with a dough hook until the dough comes together and pulls away from the sides of the bowl.

Knead the dough on medium speed for about 5 minutes until it becomes smooth and elastic, adding a sprinkle more flour only if it is excessively sticky.

Transfer the dough to a lightly oiled bowl, cover with plastic wrap or a warm towel, and let it rise in a warm, draft-free place for about 1 hour or until doubled in size.

Punch down the risen dough gently to release air, turn it out onto a floured surface, and roll it out to 1/2-inch thickness.

Use a donut cutter to cut out donut shapes and holes, placing them on a baking sheet lined with parchment paper.

Cover the shaped donuts loosely with a towel and let them rise for another 30 to 45 minutes until they look puffy and airy.

Heat about 2 inches of vegetable oil in a heavy-bottomed pot or deep fryer to 350°F (175°C), using a thermometer to maintain accuracy.

While the oil heats, whisk together the powdered sugar, milk, and vanilla extract in a medium bowl until completely smooth to create the glaze.

Carefully place 2 to 3 donuts at a time into the hot oil and fry for about 1 minute per side until they turn a light golden brown.

Remove the donuts with a slotted spoon or spider strainer and let them drain briefly on a wire rack or paper towels.

While the donuts are still warm, dip each one fully into the glaze, using a fork to flip them if necessary to coat both sides.

Place the glazed donuts back on a wire rack to allow the excess glaze to drip off and the coating to set for about 10 minutes before serving fresh.