In a large bowl, dissolve yeast in warm milk. Let it sit for about 5 minutes until it becomes foamy.
Add sugar, melted butter, egg, salt, and 2 cups of flour to the yeast mixture. Stir until just combined.
Gradually add the remaining flour until a soft dough forms. Knead the dough for about 5-10 minutes or until smooth and elastic.
Place the dough in a greased bowl, cover, and let it rise in a warm place until doubled in size, about 1 hour.
Punch down the dough and roll it out on a floured surface to about 1/2 inch thickness. Cut into donut shapes with a cutter.
Cover the cut donuts and let them rise for an additional 30 minutes.
Heat oil in a deep fryer or large heavy pot to 375°F (190°C). Fry the donuts until golden brown, turning once. Drain on paper towels.
For the glaze, whisk together powdered sugar, milk, and vanilla extract in a medium bowl until smooth. Dip warm donuts into the glaze and let it set.