In a large bowl, combine mashed potatoes, flour, sugar, baking powder, and salt.
Stir in milk, egg, and melted butter until a soft dough forms.
Roll out dough on a floured surface to 1/2 inch thickness. Cut out donuts using a donut cutter.
Heat oil in a deep-fryer or large pot to 375°F (190°C). Fry donuts until golden brown, about 1 minute per side. Drain on paper towels.
For the glaze, whisk together powdered sugar, milk, and vanilla extract until smooth. Dip warm donuts into the glaze and let excess drip off. Serve immediately.
Notes
Donuts are best enjoyed warm, fresh out of the fryer. They can be stored in an airtight container for up to 2 days.