Trim the dried base off each baby napa cabbage, halve lengthwise, quarter, then peel the layers into long strips; rinse well and drain. Slice the carrot into thin pieces and the shiitake mushrooms into slices (finely slice the stems too). Cut the white pressed tofu into thin strips.
Heat oil in a wok over medium-high heat and add the shiitake and carrot. Stir-fry until the mushrooms are fragrant and the edges just start to brown.
Add the tofu strips with a touch more oil if the pan looks dry. Pan-fry, turning occasionally, until golden brown and slightly firm on the surface.
Toss in the baby napa cabbage and turn the heat to high. Season with 1 tsp salt, 1 tbsp soy sauce and 1/2 tsp white pepper, stir-frying so every leaf is coated.
Pour in 60 ml of water and stir until the seasoning is evenly distributed. Cover the wok, lower the heat and simmer about 2 minutes until the cabbage is soft, glossy and shrunk to roughly half its volume.
Turn off the heat and drizzle in the black sesame oil. Toss once or twice to coat, then serve hot with steamed rice.