Cut the cabbage into wide, generous strips — not too fine, so it keeps its bite. Wash the cabbage, breaking up any large pieces with your hands as you go. Meanwhile, rinse the tomato and pull the shimeji mushrooms apart into loose clusters.
Heat oil in a wok or pan over medium-high heat. Add the shimeji mushrooms and toss to coat; let them sit and sear for about one minute until golden. Add the tomato and continue searing so it softens and releases its juice.
Season with the salt, sugar, and white pepper, stirring so the salt helps the tomato weep its moisture. Let the mixture cook briefly so the flavours begin to come together.
Pile the cabbage on top of the tomato-mushroom base. Cover with a lid and let it steam for about 3 minutes until the cabbage collapses and softens. You will see a pan full of clear, sweet liquid — no water added at all.
Uncover, stir everything together for ten-odd seconds so the tomato sauce fully coats the leaves. When all the flavours have melded, turn off the heat and serve immediately with steamed rice.