Preheat oven to 425 degrees. Pulse crackers, parsley, oregano, and 1 teaspoon salt in a food processor until finely ground. Season with pepper, and transfer to a plate. Puree tomatoes with juice, onion, garlic, and 1/2 teaspoon salt with an immersion blender or in clean food processor until smooth.
Coat a 9-by-13-inch baking dish with oil. Lightly whisk egg whites, and transfer to a shallow dish. Dip each chicken breast half in whites to fully coat; let excess drip off. Dredge chicken in crumb mixture, lightly pressing, to coat both sides.
Transfer chicken to prepared baking dish, and flip once to coat in oil. Bake until crisp, about 10 minutes.
Pour tomato sauce over chicken, allowing some sauce underneath each piece. Top with cheeses, and bake until bubbling and top is lightly browned, 15 to 20 minutes.