If you or your little helpers don't like spicy food, simply reduce the amount of cayenne pepper in the patties to a quarter teaspoon.
Heat broiler, with rack in highest position. In a large bowl, combine chicken, egg whites, breadcrumbs, onion, parsley, garlic, salt, and spices and mix until well blended. Line a rimmed baking sheet with foil and drizzle with 1 tablespoon oil. Divide chicken mixture into 8 portions (about 2 1/2 to 3 ounces each) and place on sheet. Shape each into an oval patty and use your hand to flatten slightly. Drizzle 1 tablespoon oil over patties.
Broil until lightly browned, about 4 minutes. Flip and cook until cooked through, about 4 minutes more. Serve in pita pockets, with lettuce, tomatoes, cucumbers, and Minted Yogurt Sauce, if desired.
In a small bowl, stir together yogurt, mint, and lime juice. Season with salt and pepper. Serve immediately or refrigerate, covered, up to 1 hour, to allow flavors to come together.