In a large bowl, beat eggs and sugar until thick. Stir in the melted butter and vanilla. Sift together the flour and baking powder, and blend into the batter until smooth.
Heat the pizzelle iron, and brush with oil. Drop about one tablespoon of batter onto each circle on the iron. You may need to experiment with the amount of batter and baking time depending on the iron. Bake for 20 to 45 seconds, or until steam is no longer coming out of the iron. Carefully remove cookies from the iron. Cool completely before storing in an airtight container.
For chocolate pizzelles, add 1/4 cup cocoa sifted together with flour and baking powder, 1/4 cup more sugar and 1/4 teaspoon more baking powder. I find that for the chocolate mixture, the iron must be well oiled to start and then brush on more as needed.