Preheat oven to 425°F. line large rimmed baking sheet with silicone baking mat or parchment paper.
In a shallow bowl, whisk together egg whites and water until foamy. Combine cheese and pepper in another shallow bowl.
Dip each green bean into egg white mixture; gently shak off any excess. Next, dip in cheese mixture and turn to coat. Place on prepared pan and repeat with remaining green beans, making sure fries do not touch or overlap; rewhip egg white mixture a few times, as needed.
Bake until cheese has melted and browned, about 8 minutes. Transfer pan to wire rack and cool several minutes.