In a medium bowl, beat the eggs with an electric mixer until light. Add the sugar, butter, lemon zest and vanilla, mix well. Stir in the flour and baking powder.
Heat pizzelle iron. Drop batter by teaspoonfuls onto the center of the patterns, close the lid and cook for about 30 seconds. Test the cooking time on the first one, because temperatures may vary. Remove cookies carefully from the iron and cool on wire racks.