In a large bowl, beat eggs and sugar until thick (pale color). 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 (here), and brush with oil. Drop about 2/3 tbsp 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 about 30 seconds, or until steam is no longer coming out of the iron. If you use the Chef's choice electric pro express bake, set to dark level #5, then bake until the light turns to green.
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.