To prepare pickled vegetables: Combine carrot, onion, vinegar, sugar and ⅛ teaspoon salt in a medium saucepan. Bring to a boil over medium-high heat and cook, stirring occasionally, until the vegetables have softened, about 5 minutes. Drain and set aside.
To prepare falafel: Place chickpeas, parsley, scallions, egg, flour, cumin and ¼ teaspoon salt in a food processor. Pulse, stopping once or twice to scrape down the sides, until a coarse mixture that holds together when pressed forms. (The mixture will be moist.) Using a generous ¼ cup each, form the chickpea mixture into 8 patties.
Heat oil in a large nonstick skillet over medium-high heat. Add the patties and cook until the bottoms are golden and beginning to crisp, 4 to 5 minutes. Carefully flip and cook until golden brown, 2 to 4 minutes more.
Meanwhile, to prepare tzatziki sauce and serve: Combine yogurt, cucumber, dill, lemon juice and salt in a medium bowl. Warm pitas, if desired. Divide the pickled vegetables and patties among the pita halves and serve with the sauce.
