Step 1: Prepare the Dough

1. In a small bowl, mix the warm milk and sugar. Sprinkle the yeast over the top and let it sit for 5-10 minutes until it becomes frothy.
2. In a large mixing bowl, combine the mashed potatoes, melted butter, eggs, and vanilla extract. Mix until smooth.
3. Add the frothy yeast mixture to the potato mixture and stir to combine.
4. Gradually add the flour and salt, mixing until a soft dough forms. Knead the dough on a floured surface for 8-10 minutes, or until it is smooth and elastic.
5. Place the dough in a lightly greased bowl, cover with a clean kitchen towel, and let it rise in a warm place for about 1 hour, or until doubled in size.

Step 2: Shape the Donuts

1. Punch down the risen dough and turn it out onto a floured surface. Roll it out to about 1/2-inch thickness.
2. Use a round cutter (3-4 inches in diameter) to cut out donut shapes. Use a smaller cutter or the end of a piping tip to cut out the centers. Re-roll the scraps to make more donuts.
3. Place the cut donuts and holes on a floured baking sheet. Cover with a kitchen towel and let them rise for 30 minutes.

Step 3: Fry the Donuts

1. Heat vegetable oil in a deep fryer or large pot to 350°F (175°C).
2. Fry the donuts in batches, about 2-3 at a time, for 1-2 minutes per side, or until golden brown. Use a slotted spoon to remove the donuts and place them on a paper towel-lined plate to drain.

Step 4: Make the Glaze

1. In a medium bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. Adjust the consistency with more milk or sugar as needed.

Step 5: Glaze the Donuts

1. Dip each donut into the glaze, ensuring it is fully coated. Place the glazed donuts on a wire rack set over a baking sheet to catch any drips. Let the glaze set before serving.