Pulse the chickpeas, parsley, onion, garlic, cumin and salt in a food processor until coarse crumbs form.
Fold in the egg and flour, shape 4 compact patties and chill them for 10 minutes.
Whisk the tahini and lemon juice until thick, then whisk in the cold water 1 tablespoon at a time until spoonable.
Coat a heavy skillet with a thin film of oil and cook the patties over medium heat for 5 to 6 minutes per side until crisp and firm.
Rest the patties for 2 minutes while the buns toast in the skillet.
Spread lemon tahini sauce over each bun and assemble with romaine, tomato and one chickpea patty.