Warm the pizzelle maker while you mix the ingredients.
In the bowl of the stand mixer (or bowl with hand mixer) beat the eggs and sugar together until the mixture is light.
Add the cold water, cooled butter, vanilla, and flour. Mix on low until the batter is smooth.
Pour 1 tablespoon of the batter onto the heated pizzelle maker and close the top. Cook according to the manufacturer’s directions, usually 30-60 seconds. Gently lift with a fork or silicone spatula and roll. Allow to cool on a wire rack. Store in a tin, not airtight, away from other cookies. They need exposure to air to stay crispy.