Drain the chickpeas, rinse in cold water, and drain well again. Place in a food processor with the tahini and egg and pulse until almost smooth but still a little coarse in texture. Transfer the mixture to a bowl and mix in the onion, carrot, cilantro, curry powder and salt. Place the breadcrumbs in a shallow dish. With hands slightly dampened with cold water, form 1/4 of the chickpea mixture (it will be quite moist) into a 3/4-inch thick patty. Coat the patty completely with the breadcrumbs, pressing on the crumbs to help them adhere. Set the coated patty on another plate. Form and coat three more chickpea burger patties. Cover and refrigerate them for 30 minutes to firm up (can be made to this point and kept refrigerated for several hours).
Heat the oil in a large non-stick skillet over medium heat. Add the burgers and cook 3 to 4 minutes per side, or until well heated through. Serve them in the buns with your choice of toppings, such as tomato, cucumber, lettuce and tzatziki sauce.