In a small bowl, combine the warmed milk and active dry yeast. Allow it to sit for about 5 minutes until it becomes frothy.

In a large mixing bowl, whisk together the flour, granulated sugar, and salt. Make a well in the center and add the melted butter and the yeast mixture.

Mix together until a dough forms. Knead the dough on a lightly floured surface for about 5-7 minutes until it becomes smooth and elastic.

Place the kneaded dough in a greased bowl, cover it with a clean cloth, and let it rise in a warm place until it doubles in size, approximately 1-2 hours.

Once risen, punch down the dough and roll it out to about 1/2 inch thick. Use a donut cutter or two round cutters (one larger and one smaller) to cut out the donuts.

Place the cut donuts on a floured baking sheet and cover them with a cloth. Let them rise again for about 30-45 minutes, or until puffy.

In a deep pot, heat the vegetable oil over medium heat until it reaches 350°F (175°C). Fry the donuts in batches, about 2-3 minutes per side, or until golden brown.

Remove the donuts from the oil and place them on a wire rack to cool slightly.

For the glaze, whisk together the powdered sugar, whole milk, and vanilla extract in a bowl until smooth.

Dip the warm donuts into the glaze, allowing any excess to drip off before placing them back on the wire rack to set.