Place the chickpeas, egg, garlic, salt, cumin, cayenne, lemon zest, flour and parsley in food processor. Pulse until a coarse mixture forms that holds together. It should be moist and sticky.
Place chickpea mixture in a bowl and mix in grated carrot and red onion. Take a ½ cup of the mixture and firmly press it into a patty about 3 ½ inches in diameter and ¼ inch thick. You can also form these into mini-patties for appetizers. Place on a plate or a baking sheet and refrigerate for at least 30 minutes and up to overnight. Cover if refrigerating for more than a few hours.
Heat a couple tablespoons of oil or ghee in a large skillet over medium heat until hot, but not smoking. Add patties to the skillet (do this in batches, if necessary) and cook until crisp and golden brown on the underside, about 6 minutes. Carefully turn over and cook until golden brown on the other side, about 3-5 minutes more.
Serve with or without a warmed pita half or a hamburger bun and suggested toppings.