Take the bulgur in a mixing bowl and mix with 1/4 teaspoon of salt. Boil a cup of water and pour it over the bulgur. Cover the bowl and let the water soften the wheat. This will take around half an hour. Drain off the water in a sieve when the wheat goes tender. However, it should still have its chewy texture. Press the wheat in the sieve to get rid of all excess water.
Place the beans in another mixing bowl while the bulgur is softening up. Mash them into a paste form using a masher. Add paste the breadcrumbs, scallions, carrot, cayenne, egg, tahini, and bulgur. Season with salt and pepper according to taste. Use freshly ground pepper if possible. Combine everything well together so that a smooth mixture is formed.
Make four patties out of this paste. Warm oil in a skillet and roast the patties on each side until they are brown and cooked. Give each side 5 to 8 minutes. Serve the patties on the muffins. Garnish with mayonnaise and lettuce leaf. A wonderful treat for the four people who feast on these burgers.