In a mixing bowl, combine warm milk, sugar, and yeast. Let sit for 5 minutes until frothy.
Add egg, melted butter, vanilla, and gradually mix in flour to form a soft dough.
Knead the dough on a floured surface until smooth. Place in a greased bowl, cover, and let rise for 1 hour.
Roll out dough to 1/2 inch thickness. Cut out doughnuts with a doughnut cutter.
Heat oil in a deep fryer or large pot to 375°F (190°C).
Fry doughnuts in batches until golden brown. Drain on paper towels.
For the glaze, whisk together powdered sugar, milk, and vanilla until smooth.
Dip each doughnut into the glaze, then let set on a wire rack.
Prep Time: 1 hour 30 minutes | Cooking Time: 20
