In a mixing bowl, whisk together the flour, sugar, baking powder, and salt.
In another bowl, beat the eggs and add the milk, melted butter, and vanilla extract. Mix well.
Pour the wet ingredients into the dry ingredients and stir until a soft dough forms.
On a floured surface, roll out the dough to about ½-inch thickness.
Use a donut cutter or a round cookie cutter to cut out donuts. You can also use a small round cutter for the holes.
In a deep pan, heat about 2 inches of vegetable oil to 350°F (175°C).
Carefully place the donuts into the hot oil. Fry each side for about 1-2 minutes or until golden brown.