In a large bowl, whisk together egg, chickpea flour, water, soy sauce and garlic. Add onion, cabbage, mushrooms and sesame seeds; season with 1 teaspoon each salt and pepper. Using your hands, massage the cabbage mixture to evenly coat with the chickpea batter. Cover and let the mixture sit at room temperature for 30 minutes. Juices from the vegetables will leach out and make the batter slightly wetter.
Divide the cabbage mixture into 6 portions (about 1½ cups each).
Heat ¼ cup vegetable oil in a large cast-iron pan over medium-low heat. Cook 3 portions at a time, coaxing each patty together in the pan and compressing them with a flat spatula. Gently fry the patties, without disturbing them, until one side is golden brown, 7 to 8 minutes. Flip and cook the other side until the golden brown and the vegetables are cooked through, 5 to 7 minutes. Remove the patties from the pan onto a wire rack. Add remaining ¼ cup oil to the pan and cook the remaining 3 portions.
Place a slice of cheese on each cooked vegetable patty and allow the residual heat from the patties to melt the cheese. Serve sandwiched between buns, topping with mayonnaise, tomatoes, pickles and lettuce.