Combine eggs, sugar, vegetable oil, butter, and anise oil in the bowl of a stand mixer fitted with a paddle attachment. Beat until creamy and well combined. Mix in 1 cup flour, baking powder, and salt until combined. Slowly add remaining flour, 1 cup at a time, mixing on low speed until mixture resembles a thick pancake batter.

Preheat a pizzelle cooking iron according to manufacturer's instructions. Drop a teaspoon of dough onto each plate of the cooking iron and close the lid. Cook just until steaming slows, 10 to 20 seconds. Repeat with remaining batches of batter.