1. Slice the potatoes lengthwise into matchsticks, about 1/4 inch thick. In a large bowl, combine the water, vinegar, and potatoes. Set aside, uncovered, for 15 minutes at room temperature.
2. Preheat an oven to 400 degrees F. Coat a large baking sheet with nonstick cooking spray. Place the potatoes on the prepared baking sheet in a single layer. In a small bowl, using an electric mixer on high speed, beat the egg whites until foamy. Brush the tops and sides of the potatoes with the beaten egg whites and sprinkle with the paprika and salt.
3. Bake, turning several times, until golden brown and crisp, about 40 minutes. To serve, divide among 4 individual plates.