In a large bowl, whisk together the warm milk and yeast; let it sit for 5 minutes until foamy.

Add the sugar, egg, melted butter, and salt to the yeast mixture and whisk until well combined.

Gradually add the flour, mixing until a soft, shaggy dough forms, then knead on a lightly floured surface for 5-7 minutes until smooth and elastic.

Place the dough in a lightly oiled bowl, cover, and let it rise in a warm place for about 30 minutes, or until doubled in size.

Punch down the dough, then roll it to 1/2-inch thickness and cut out small circles using a round cutter or the wide end of a piping tip.

Heat vegetable oil in a deep pot or Dutch oven to 350°F (175°C), then carefully fry the doughnut holes in batches for 2-3 minutes until golden brown and cooked through.

Remove the fried doughnut holes with a slotted spoon and place them on a wire rack lined with paper towels to drain excess oil.

While the doughnut holes are cooling slightly, whisk together the powdered sugar, milk, and vanilla extract in a shallow bowl until smooth.

Dip each warm doughnut hole into the glaze, turning to coat completely, then place back on the wire rack to allow the glaze to set.