In a small bowl, combine warm water and yeast. Let it sit for about 5 minutes until it becomes frothy.

In a large mixing bowl, whisk together sugar, salt, and 2 cups of flour.

Add the softened butter, warm milk, egg, and yeast mixture into the flour mixture. Mix until a soft dough forms.

Turn the dough onto a floured surface and knead for about 5-7 minutes until smooth and elastic.

Place the dough in a greased bowl, cover with a damp cloth, and let it rise in a warm place for about 1 hour or until it doubles in size.

Once risen, roll out the dough on a floured surface to about 1/2 inch thickness.

Use a donut cutter or two round cutters (one large and one small) to cut out donut shapes.

Place the donuts on a baking sheet lined with parchment paper, cover, and let them rise for another 30 minutes.

Heat vegetable oil in a deep fryer or a heavy-bottomed pot to 350°F (175°C). Carefully fry the donuts a few at a time until golden brown, about 1-2 minutes per side. Drain on paper towels.

For the glaze, whisk together powdered sugar, milk, and vanilla extract until smooth.

Dip the warm donuts into the glaze, then place on a wire rack to let the glaze set.