In a large wok or skillet, heat a slick of oil over medium-high heat until very hot. Add the sliced peppers and sliced onions and cook, stirring regularly, until beginning to char around the edges, approximately 5 minutes. Season with salt and transfer to a bowl. Once in the bowl, season generously with shichimi togarashi to taste.
Next, cook the mushrooms. If the skillet seems dry, add another slick of oil. Once hot, add the mushrooms. Sauté until golden brown and beginning to crisp up around the edges. Transfer to the bowl of peppers and sprinkle with more shichimi.
Finally, cook the cabbage. Again, if the skillet seems dry, add more oil. Once very hot, add the cabbage and cook, stirring regularly, until it begins to char and turn golden-brown, about 5-6 minutes. Season with salt and transfer to the bowl of warm vegetables and sprinkle with shichimi, if desired.