In a medium bowl stir together flour, baking powder, and salt; set aside. In a large bowl whisk eggs until frothy. Gradually add granulated sugar, whisking constantly. Stir in melted butter and vanilla. Fold in flour mixture. Set batter aside.
Lightly coat an electric pizzelle iron with oil; heat according to the manufacturer's directions. (Or coat a nonelectric pizzelle iron with oil; heat on range top over medium heat until a drop of water sizzles on the surface. Reduce heat to medium-low.)
For each pizzelle, place 1 slightly rounded tablespoon of batter on pizzelle iron, slightly off-center toward the back. Close lid; bake according to the manufacturer's directions. (If using a nonelectric iron, bake about 2 minutes or until light brown, turning once.) Transfer warm pizzelles to a wire rack; cool. If desired, sprinkle pizzelles with powdered sugar.