Preheat a 2-cavity waffle iron for about 5 minutes or until hot.
While the waffle iron is preheating, lightly whisk the eggs and vanilla extract in a medium bowl. Using a fork or a rubber spatula, mix in the shredded mozzarella.
Brush the two waffle iron cavities with 1 tablespoon of melted butter.
Pour ¼ of the mixture (about ½ cup) into each cavity.
Cook the chaffles until browned and crispy, 5-7 minutes. Start checking after 5 minutes.
Remove the cooked chaffles to a baking dish and place them in the oven to keep them warm.
Brush the waffle iron cavities with the remaining butter. Give the remaining mixture another stir, then pour it into the waffle iron's cavities, dividing it evenly. Cook until browned, about 5-7 minutes.
I don't add any sweetener because I drizzle the chaffles with sugar-free maple syrup.
Chaffles tend to stick to waffle makers more than other waffles, so make sure your waffle iron is truly nonstick and generously grease it. If the chaffles stick, gently release them from the waffle iron by sliding a cake server or a dinner knife around their edges, then sliding a wide spatula underneath them.
You can keep the leftovers in the fridge, in an airtight container, for up to 4 days. Reheat them in a toaster on the lowest setting. You can also freeze chaffles in a freezer bag. Separate them with wax paper squares so they don't stick together.