In a small bowl, dissolve yeast in warm water and let sit for 5 minutes until foamy.
In a large bowl, mix warm milk, sugar, salt, egg, and melted butter. Add the yeast mixture.
Gradually mix in flour until a soft dough forms. Knead on a floured surface for 5 minutes.
Place the dough in a greased bowl, cover, and let rise for 1 hour or until doubled in size.
Roll the dough to 1/2-inch thickness and cut out donut shapes. Cover and let rise for 30 minutes.
Heat oil in a deep skillet to 375°F. Fry donuts for 1-2 minutes on each side until golden brown. Drain on paper towels.
For the glaze, whisk powdered sugar, milk, and vanilla in a bowl until smooth.
Dip warm donuts into the glaze and place on a wire rack to set.