Begin by preheating your pizzelle maker according to the manufacturer’s guidelines. Meanwhile, prepare the batter.

In the bowl of a stand mixer—or using a handheld mixer—beat together the eggs and granulated sugar until the mixture becomes pale and slightly thickened, a sign that air has been incorporated for a lighter texture.

Gradually add the cold water, followed by the cooled melted butter and vanilla extract, mixing gently to combine. Sift in the flour and continue mixing on a low speed until the batter is silky and free of lumps.

Once the pizzelle maker is fully heated, drop approximately 1 tablespoon of batter onto the center of the hot plate. Close the lid and cook following the manufacturer's timing instructions, generally between 30 to 60 seconds, or until the wafer is golden and crisp.

Carefully lift the wafer using a fork or a silicone spatula to avoid breaking its delicate structure. While still warm and pliable, roll the wafer into a tight cylinder and place it on a wire rack to cool completely, where it will crisp up beautifully.

For best results, store the finished wafer rolls in a traditional tin container. Avoid airtight storage and keep them separate from other cookies; a slight exposure to air ensures they maintain their irresistible crispness.