Preheat the oven to 425°F. Coat 2 rimmed baking sheets with cooking spray.
Separate the onion into rings and place in a large bowl. Add enough water to cover.
Combine the flour and ½ teaspoon each salt and pepper on a shallow plate or dish. Whisk together the egg whites, mayonnaise, and ⅓ cup water in a small bowl. Place the panko on a separate shallow plate or dish.
Working 1 onion ring at a time, dredge the onion in the flour, turning to coat. Shake off excess and dip in the mayonnaise mixture. (I use a fork to do this.) Coat in the panko and place on the baking sheet. Repeat the process until all the onion rings are coated. Coat with cooking spray.
Bake the onion rings at 425°F for 15 minutes. Turn them over and bake for another 10 minute or until they are browned and crispy. Sprinkle with additional salt while still hot.