Prepare the Dough: In a large mixing bowl, combine the flour, sugar, salt, and nutmeg. Whisk together to blend the dry ingredients.

In a separate bowl, whisk the egg with the warm milk and melted butter until well combined.

Sprinkle the dry yeast over the wet ingredients and let it sit for 5 minutes to activate.

Gradually add the wet mixture to the dry ingredients, stirring until a soft dough forms.

Knead the dough on a lightly floured surface for about 5-7 minutes, until it becomes smooth and elastic.

Place the dough in a lightly greased bowl, cover with a clean towel, and let it rise in a warm place for 1-2 hours, or until doubled in size.

Shape the Donuts: Once the dough has risen, punch it down to release the air.

Roll the dough out on a floured surface to about 1/2 inch thickness.

Use a donut cutter or two round cutters (one large, one small) to cut out donut shapes. Save the donut holes for frying as well.

Place the donuts on a parchment-lined baking sheet, cover with a towel, and let them rise again for about 30 minutes, or until puffy.

Fry the Donuts: Heat oil in a deep fryer or large pot to 350°F (175°C).

Carefully drop the donuts into the hot oil, frying them in batches to avoid overcrowding.

Fry each side for about 1-2 minutes, or until golden brown.

Remove the donuts with a slotted spoon and place them on a paper towel-lined plate to drain excess oil.

Make the Glaze: In a medium saucepan, melt the butter over low heat.

Remove from heat and whisk in the powdered sugar, milk, and vanilla extract until smooth.

If the glaze is too thick, add more milk, one tablespoon at a time, until you reach your desired consistency.

Glaze the Donuts: Dip each warm donut into the glaze, making sure both sides are coated.

Place the glazed donuts on a wire rack to allow the glaze to set. For a thicker glaze, you can dip the donuts a second time.

Serve and Enjoy: Serve the glazed donuts warm for the best flavor and texture. They are perfect on their own, or with a cup of coffee or tea.