In a large bowl, combine warm milk and yeast. Let sit until foamy, about 5 minutes.
Add sugar, salt, eggs, and melted butter to the yeast mixture. Mix well.
Gradually add flour, mixing until dough pulls away from the sides of the bowl.
Knead dough until smooth and elastic, about 5-7 minutes.
Place in an oiled bowl, cover, and let rise in a warm place until doubled in size, about 1 hour.
Punch down dough and roll out to ½-inch thickness on a floured surface.
Cut with a doughnut cutter and place on floured baking sheets.
Cover and let rise again until puffy, about 30-45 minutes.
Heat oil to 375°F (190°C) and fry doughnuts until golden brown, about 1 minute per side.
Drain on paper towels before glazing or filling.