In a mixing bowl, combine flour, sugar, baking powder, and salt.
In another bowl, whisk together milk, melted butter, vanilla extract, and eggs.
Gradually add the wet ingredients to the dry ingredients, mixing until a dough forms.
Roll out the dough on a floured surface to about ½ inch thickness.
Cut out doughnuts using a doughnut cutter.
Heat oil in a frying pan to 375°F (190°C).
Fry the doughnuts until golden brown, about 1-2 minutes per side.
Remove and drain on paper towels. Let cool slightly before serving.