Preheat the oven to 425 degrees F. Coat 2 baking sheets with cooking spray. Whisk the egg whites, paprika and 1/2 teaspoon each salt and pepper in a large bowl. Add the potatoes; 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, 20 to 30 minutes.
Meanwhile, coat a large nonstick skillet with cooking spray. Add half of the scallions; cook over medium heat, stirring, until softened, 1 to 2 minutes. Add 1/2 cup chicken broth. 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, 6 to 8 minutes. Stir in the chicken and peas and warm through, about 3 minutes. (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.