In a mixing bowl, whisk together the flour, sugar, baking powder, salt, and nutmeg.

In another bowl, beat the egg and then add the milk, melted butter, and vanilla extract.

Combine the wet ingredients with the dry ingredients and mix until just combined.

Heat the vegetable oil in a frying pan over medium heat.

Using a cookie cutter, cut out doughnut shapes from the batter.

Fry the doughnuts in the hot oil until golden brown, about 2-3 minutes on each side.

Remove the doughnuts with a slotted spoon and drain on paper towels.

Dust the warm doughnuts with powdered sugar before serving.