To prepare the tofu, in a large bowl, whisk the salt into the water until dissolved. Gently add the tofu blocks and soak for about 10 minutes. Remove the tofu and pat dry thoroughly with paper towels.
To make the sauce, in a bowl, whisk together the water, soy sauce, dark soy sauce, and vegetarian oyster sauce until smooth. Set aside.
To pan-fry the tofu, in a nonstick skillet over medium heat, add 3 tablespoons of cooking oil. Carefully add the tofu pieces and pan-fry for about 3 minutes per side until golden brown.
To stir-fry the vegetables, in a wok over high heat, add the remaining cooking oil. Stir-fry the carrot, ginger, garlic, shiitake mushrooms, and white beech mushrooms for about 30 seconds until fragrant.
To combine and thicken, pour the sauce into the wok and bring to a boil. Return the tofu to the wok and gently toss to coat evenly. In a small bowl, combine the cornstarch and 1 ½ tablespoons water to make a slurry. Stir the slurry into the wok and cook until the sauce thickens.
To serve, transfer everything to a preheated clay pot and top with the scallions. Serve hot.