1. In a small bowl, dissolve yeast in warm water. Let stand until frothy, about 5 minutes.
2. In a large mixing bowl, combine the yeast mixture, milk, sugar, melted butter, beaten eggs, flour, and salt. Mix until a dough forms.
3. Knead the dough on a floured surface until smooth and elastic, about 5 minutes. Place in a greased bowl, cover, and let rise in a warm place until doubled in size, about 1 hour.
4. Roll out the dough to ½ inch thickness. Cut with a floured doughnut cutter. Let doughnuts rise again until doubled in size, about 30 minutes.
5. Heat oil in a deep fryer or large pot to 350°F (175°C). Fry doughnuts in hot oil until golden brown, turning once. Drain on paper towels.