In a food processor combine the beans, carrot, flour, parsley, 1 tablespoon olive oil, halved garlic, coriander, cumin, salt, and black pepper. Cover and process until finely chopped and mixture holds together (should have some visible pieces of garbanzo beans and carrots).
Shape mixture into four 3-inch patties. Heat 2 tablespoons oil in a large skillet over medium-high heat. Add patties. Cook for 2 to 3 minutes per side or until browned and heated through.
Meanwhile, in a small bowl stir together mayonnaise, minced garlic, and cayenne pepper. Spread mixture over pita rounds. Top with spinach and falafels.