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

In another bowl, combine the milk, egg, melted butter, and vanilla extract.

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

Turn the dough onto a floured surface and roll out to about 1/2 inch thick.

Use a cookie cutter to cut out doughnut shapes.

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

Carefully place the doughnuts in the hot oil and fry until golden brown, about 2-3 minutes per side.

Remove and drain on paper towels.

Dust the warm doughnuts with powdered sugar before serving.