In a bowl, combine warm milk and sugar. Sprinkle yeast on top and let sit 5–10 minutes until foamy.
Add eggs and melted butter to yeast mixture. Gradually mix in flour and salt until dough forms.
Transfer to a floured surface and knead 6–8 minutes until smooth and elastic.
Place dough in a greased bowl, cover, and let rise 1–1½ hours until doubled in size.
Roll dough to ½-inch thickness. Cut into donut shapes using cutter.
Place donuts on parchment-lined tray. Cover and let rise 30–45 minutes.
Heat oil to 350°F (175°C). Fry donuts 1–2 minutes per side until golden brown. Drain on paper towels.
Whisk powdered sugar, milk, and vanilla until smooth.
Dip warm donuts into glaze. Place on wire rack to set.