Whisk the egg whites, paprika and 1/2 teaspoon each salt and pepper in a large bowl.
Add the potatoes and toss to coat.
Let the excess egg drip off; then spread out on the baking sheets. Mist with cooking spray and bake, flipping halfway through, until golden and crisp, about 20 to 30 minutes.
Meanwhile, coat a large nonstick skillet with cooking spray.
Add half of the scallions and cook over medium heat, stirring, until softened, about 1 to 2 minutes.
Add 1/2 cup chicken broth and whisk in the flour until smooth, then gradually whisk in the remaining 2 1/2 cups broth and the Worcestershire sauce. Simmer, whisking occasionally, until thickened, about 6 to 8 minutes.
Now stir in the chicken and peas and warm through, about 3 minutes. (You can add up to 1 cup water if the gravy is too thick.)
Scatter the mozzarella evenly over the chicken mixture; layer the fries and chicken mixture in bowls; top with the remaining scallions.