1. In a mixing bowl, combine the warm milk, sugar, and yeast. Let it sit for about 5 minutes, until it becomes frothy.
2. Add the beaten egg, melted butter, and vanilla extract to the yeast mixture. Gradually mix in the flour until a soft dough forms.
3. Transfer the dough to a floured surface and knead until smooth. Place the dough in a greased bowl, cover, and let it rise for 1 hour.
4. Roll out the dough to a thickness of about 1/2 inch. Use a doughnut cutter to cut out doughnut shapes.
5. Heat oil in a deep fryer or large pot to 375°F (190°C).
6. Fry the doughnuts in batches until golden brown. Remove and drain on paper towels.
7. For the glaze, whisk together powdered sugar, milk, and vanilla extract until smooth.
8. Dip each doughnut into the glaze and allow to set on a wire rack.

Prep Time: 1 hour 30 minutes
Cook Time: 20 minutes
Total Time: 1 hour 50 minutes
Servings: 12 doughnuts
Calories: 210 kcal per doughnut