In a mixing bowl, combine flour, sugar, baking powder, salt, and nutmeg.

In another bowl, whisk together the egg, milk, and melted butter.

Pour the wet ingredients into the dry ingredients and mix until just combined.

Knead the dough on a floured surface for a few minutes until smooth.

Roll out the dough to about ½ inch thick and cut out doughnuts using a cookie cutter.

Heat vegetable oil in a frying pan to 350°F (175°C).

Fry the doughnuts in batches for about 2-3 minutes on each side until golden brown.

Remove from oil and drain on paper towels.

In a small bowl, whisk together powdered sugar, milk, and vanilla extract to make the glaze.

Dip the warm doughnuts into the glaze and let them set on a wire rack.