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

In another bowl, mix the warm milk, eggs, melted butter, and vanilla extract.

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

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

Use a cookie cutter to cut out doughnut shapes and place them on a floured tray.

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

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

Remove the doughnuts and place them on a cooling rack.

Dust the warm doughnuts with powdered sugar before serving.