Heat cooking oil in a claypot or deep pot over medium heat. Sauté minced garlic and ginger until fragrant.
Add sliced shiitake mushrooms and carrots. Stir-fry for 3-5 minutes until slightly softened.
Pour in mushroom broth, vegetarian oyster sauce, soy sauce, dark soy sauce, sugar, and white pepper. Bring to a simmer.
Gently add tofu cubes to the simmering sauce. Cover and braise for 10-15 minutes, allowing the tofu to absorb the flavors.
Add the soaked and drained glass noodles to the claypot. Stir gently to mix. Cook for another 3-5 minutes until the noodles are tender and have absorbed most of the liquid.
Stir in the cornstarch slurry to thicken the sauce to your desired consistency. Drizzle with sesame oil before serving.
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