Warm milk to about 105°F. Stir in sugar and yeast. Let sit 5 minutes.
Add eggs and melted butter.
Slowly mix in flour and salt until a dough forms. Knead/mix for about 5 minutes.
Let dough rest 10 minutes, then transfer to a greased bowl, cover, and refrigerate at least 2 hours or overnight.
Roll dough out on a floured surface to about ½-inch thick. Cut donuts using a donut cutter.
Place donuts on a greased baking sheet, cover, and let rise about 1 hour until puffy.
Heat oil to 350–375°F. Fry donuts about 90 seconds per side until golden brown. Donut holes cook faster.
Remove to paper towels briefly.
For the glaze, whisk together melted butter, vanilla, powdered sugar, milk, and salt until smooth.
Dip warm donuts into glaze and enjoy!