In a large bowl, whisk together the flour, sugar, baking powder, salt, and ground nutmeg (if using).
In a separate bowl, beat the eggs and then add the milk and vanilla extract. Mix well.
Gradually add the wet ingredients to the dry ingredients, mixing until a smooth dough forms.
On a lightly floured surface, roll out the dough to about 1/2 inch (1.3 cm) thickness.
Use a doughnut cutter or two round cutters (one larger and one smaller) to cut out doughnut shapes. If you don’t have a cutter, you can use a glass for the large circles and a bottle cap for the smaller ones.
In a deep frying pan or pot, heat the vegetable oil over medium heat until it reaches 350°F (175°C). If you don’t have a thermometer, you can test the oil by dropping a small piece of dough into it; if it sizzles and rises to the surface, the oil is ready.
Carefully place a few doughnuts into the hot oil, being careful not to overcrowd the pan.
Fry the doughnuts for about 2-3 minutes per side, or until they are golden brown.
Use a slotted spoon to remove the doughnuts from the oil and place them on a paper towel to drain excess oil.
Dust the doughnuts with powdered sugar if desired.
Enjoy your meal!