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In a large bowl, combine warm milk, sugar, and yeast. Let it sit for 5 minutes to activate the yeast.
Add the flour, egg, melted butter, and salt to the mixture. Stir until a dough forms.
Knead the dough on a floured surface for about 5 minutes until smooth. Place it in a greased bowl, cover with a cloth, and let it rise for 1 hour.
Roll out the dough on a floured surface to about ½ inch thickness. Use a donut cutter to shape the donuts.
Heat vegetable oil in a large pot to 375°F (190°C). Fry the donuts for 1-2 minutes on each side until golden brown.
Remove the donuts from the oil and let them cool on a wire rack.
For the glaze, whisk together powdered sugar, milk, and vanilla extract until smooth.
Dip each donut into the glaze while it's still warm, then place it back on the rack to set.
Serving and Storage Tips:
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