To make the sauce, in a bowl, mix the vegetable stock, soy sauce, dark soy sauce, and oyster sauce.
To prepare the tofu, cut the egg tofu into rings and gently coat each piece with the cornstarch.
To fry the tofu, in a pan over high heat, heat the cooking oil to about 350°F (177°C), then add the tofu and fry for about 3 minutes until lightly golden, then remove and drain.
To stir-fry the aromatics, in a wok over high heat, add the cooking oil and stir-fry the carrot, ginger, garlic, shiitake mushrooms, and beech mushrooms for about 30 seconds until fragrant.
To braise, pour in the sauce and return the tofu, bring to a simmer, then add the cornstarch mixed with water and stir gently until the sauce thickens.
To serve, transfer to a clay pot if using, top with the scallions, and serve hot.