Preheat a pizzelle iron. Lightly grease if your plates aren’t nonstick.
Whisk the beaten eggs and sugar in a bowl until pale and slightly thick, about 2 minutes.
Stream in the melted, cooled butter while whisking until smooth.
Stir in the anise extract, vanilla, and lemon zest.
In a separate bowl, whisk the flour, baking powder, and salt together.
Fold the dry ingredients into the wet mixture just until no dry spots remain. The batter will be thick.
Let the batter rest for 5 minutes while the iron fully heats.
Spoon about 1 tbsp batter onto the iron per cookie. Close and cook until pale golden, 45–60 seconds.
Lift cookies with a thin spatula and cool on a wire rack to crisp. Repeat with remaining batter.
Dust cooled pizzelle with powdered sugar before serving.