To prepare the tofu, on a cutting board, cut the tofu into small, even pieces, then sprinkle with the salt and let rest for 5 minutes. Gently pat the tofu dry with paper towels to remove excess moisture.
To make the sauce, in a bowl, combine the water, soy sauce, dark soy sauce, and vegetarian oyster sauce, then mix until smooth.
To pan-fry the tofu, in a pan over medium heat, heat 3 tablespoons of the cooking oil until hot, then carefully add the tofu and pan-fry for about 3 minutes per side until lightly golden and set. Remove the tofu.
To build the base, in a wok over high heat, add the remaining 1 tablespoon cooking oil, then add the carrot, ginger, garlic, king oyster mushrooms, shiitake mushrooms, beech mushrooms, and snap peas, and stir-fry for about 30 seconds until fragrant.
To braise, pour in the sauce and bring to a simmer. In a small bowl, mix the cornstarch with the water to form a slurry, stir again, then add to the wok and cook until the sauce thickens. Return the tofu and gently toss to coat.
To serve, transfer everything to a preheated clay pot, top with the scallions, and serve immediately.