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

In another bowl, beat the egg and then mix in the milk, melted butter, and vanilla extract.

Combine the wet ingredients with the dry ingredients until just mixed.

Knead the dough gently on a floured surface and roll it out to about 1/2 inch thick.

Use a cookie cutter to cut out doughnut shapes.

Heat the vegetable oil in a frying pan over medium heat.

Carefully place the doughnuts in the hot oil, frying them for about 2-3 minutes on each side until golden brown.

Remove the doughnuts with a slotted spoon and place them on paper towels to drain excess oil.

Dust the warm doughnuts with powdered sugar before serving.

Enjoy your delicious homemade doughnuts!