Whisk bibimbap sauce ingredients together in a small bowl. Set aside.
Slice tofu into 8-12 slabs, each between ¼ and ½ inch thick. Any shape will do, as long as they're sized to fit your rolls. Set aside.
Coat a large, flat bottomed skillet with 1 tablespoon of oil and place over medium heat. Arrange shiitake caps in skillet and cover. Allow to cook for about 4 minutes, until lightly browned and tender. Remove from skillet and transfer to plate.
Add carrots and garlic to skillet. Stir-fry until tender-crisp, about 2 minutes. Remove from skillet and transfer to a plate.
Recoat skillet with remaining 1 tablespoon of oil. Arrange tofu slabs in skillet and cook until browned on bottoms, about 5 minutes. Flip and cook another 5 minutes on opposite sides. Flip again, reduce heat to low and spoon about half of your bibimbap sauce over tofu. Cook about 1 minute, flip and cook another minute, until sauce coats tofu and is lightly caramelized in spots.
Slather sandwich rolls with mayo and stuff with tofu, shiitakes, carrots, spinach and kimchi. Drizzle with additional sauce and sprinkle with sesame seeds and scallions.