1. In a small bowl, dissolve yeast in warm water. Let it stand until frothy, about 5 minutes.
2. In a large mixing bowl, combine the yeast mixture, warm milk, sugar, salt, eggs, butter, and 2 cups of flour. Beat on medium speed until smooth. Stir in remaining flour to form a soft dough.
3. Turn dough onto a floured surface; knead until smooth and elastic, about 6-8 minutes. Place in a greased bowl, turning once to grease the top. Cover and let rise in a warm place until doubled, about 1 hour.
4. Punch dough down. Turn onto a lightly floured surface; roll to ½-in. thickness. Cut with a floured donut cutter. Place on greased baking sheets. Cover and let rise until doubled, about 30 minutes.
5. In an electric skillet or deep fryer, heat oil to 375°F. Fry donuts, a few at a time, until golden brown on both sides. Drain on paper towels.
6. For glaze, in a large bowl, combine powdered sugar, milk, and vanilla until smooth. Dip warm donuts in glaze. Cool on wire racks.