In a medium bowl combine flour, baking powder, and salt; set aside. In a large mixing bowl combine eggs, sugar, and vanilla. Beat with an electric mixer on medium speed for 3 minutes or until thick. In a small bowl combine milk and melted butter.Add the flour mixture and milk mixture alternately to the egg mixture, beating on low speed after each addition just until combined. Cover and chill the dough for 2 to 4 hours.

About a half-hour before the end of the chilling time, get the coating, glaze, or icing ready for the doughnuts. Coatings and toppings should be prepared before you start to fry the doughnuts, as they should be applied while the fried doughnuts are still slightly warm. Doughnuts can simply be coated with powdered sugar or granulated sugar. Place the sugar in a shallow dish, such as a pie plate. Or you can ice the tops of the doughnuts

On a well-floured surface, roll the dough to 1/2-inch thickness. Do not stir additional flour into the dough; this can cause the finished product to be heavy and dry.

Use a floured 2-1/2-inch doughnut cutter to cut the dough into rings. Dip the cutter into flour between cuts to prevent the dough from sticking to the cutter. Each cut should produce one doughnut and one doughnut hole.Reroll dough scraps as necessary to make about 16 doughnuts and 16 doughnut holes.

If using an electric deep-fat fryer, heat the oil to 365°F according to manufacturer's directions. Or, in a heavy, deep large saucepan, heat the oil to 365°F.

Add these optional finishing touches while the doughnuts are still slightly warm: To coat the doughnuts with powdered sugar or granulated sugar, simply roll them in the sugar until coated on all sides.To glaze, dip the tops of the doughnuts in the Chocolate Glaze, Powdered Sugar Icing, or Chocolate Powdered Sugar Icing (see Step 3, above). If desired, sprinkle iced doughnuts with chopped nuts, flaked coconut, or candy sprinkles. Allow the doughnuts to dry on a wire cooling rack.