In a medium mixing bowl, use an electric hand mixer to beat eggs and sweetener on medium-high speed until yellow, light, and bubbly, about 2-3 minutes.
Melt coconut oil in the microwave for about 30 seconds. Allow to cool slightly.
Add coconut oil and anise extract to egg mixture. Beat on medium speed until incorporated.
In a small mixing bowl, sift together coconut flour, baking powder, and salt.
Add coconut flour (dry) mixture to egg (wet) mixture. Combine on low speed until a moist dough forms. (See Note 1)
Plug in pizzelle iron and allow it to heat according to the manufacturer's instructions. Grease slightly with leftover coconut oil or butter.
Using a medium cookie scoop (about 2 tablespoons in size) or a tablespoon, form about 10-12 dough balls. Place 1 dough ball in the pizzelle maker slightly above the floral pattern, flatten slightly before closing the lid. Press down on the pizzelle maker lid until completely flattened. Peek inside to make sure it’s completely covering the design. There may be a little spillage over the sides.
Cook until lightly golden brown, usually 2-3 minutes. Lift the lid to check doneness.
Remove from pizzelle maker with a small spatula and place on a wire cooling rack. Allow to cool completely. (See Note 3)
Sprinkle with powdered sugar-free sweetener if desired before serving.