Heat the olive oil a large, deep skillet or Dutch oven (3.5 quarts or larger) over medium-high heat and sauté the onion, carrots and celery until they start to soften, about 5 minutes.
Add in the shredded cabbage, mushrooms, tamari, salt and pepper, along with a splash of water to help prevent sticking. The water should immediately sizzle when you add it to the pan, creating steam.
Cover the skillet and lower the heat, cooking until the vegetables are tender, about 10 to 15 minutes depending on how crunchy or soft you want the vegetables to be. Stir in the sesame oil as soon as the vegetables are tender, so that it's not exposed to too much heat.
If you'd like to add in some eggs or tofu to this dish, move the cooked vegetables to the edge of the pan to create a well in the center. Add a touch more oil and scramble the protein directly in the pan until cooked through. Stir to combine with the vegetables.
Season with additional salt and pepper, if desired, then serve warm with a topping of green onions and sesame seeds.