Step 1: Prepare the Dough

1. In a large bowl, whisk together the flour, baking powder, salt, nutmeg, and sugar.
2. In a separate bowl, whisk together the melted butter, milk, vanilla extract, and egg.
3. Gradually add the wet ingredients to the dry ingredients, stirring until a soft dough forms. Be careful not to overmix, as this can make the donut holes dense.

Step 2: Heat the Oil

1. In a deep, heavy-bottomed pot, heat the vegetable oil to 350°F (175°C). Use a candy or frying thermometer to maintain the correct temperature.
2. While the oil heats, prepare a plate lined with paper towels to drain the fried donut holes.

Step 3: Shape and Fry the Donut Holes

1. Using a small cookie scoop or a teaspoon, scoop out portions of dough and roll them into small balls with your hands.
2. Carefully drop a few dough balls at a time into the hot oil, being careful not to overcrowd the pan.
3. Fry for 2-3 minutes, turning occasionally, until golden brown on all sides.
4. Remove with a slotted spoon and transfer to the paper towel-lined plate to drain excess oil.
5. Repeat with the remaining dough, ensuring the oil stays at the right temperature.

Step 4: Make the Glaze

1. In a medium bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.
2. If the glaze is too thick, add a little more milk. If it’s too thin, add more powdered sugar.

Step 5: Glaze the Donut Holes

1. While the donut holes are still warm, dip each one into the glaze, rolling it around to coat evenly.
2. Place them on a wire rack over a baking sheet to allow excess glaze to drip off.
3. Let the glaze set for about 10 minutes before serving.