Heat a large pot of water to boiling and salt it generously. Cook the orzo until barely al dente, about 6 to 7 minutes. Drain and toss with a generous drizzle of oil so that the grains of orzo are lightly coated with oil; set aside.
Heat a large sauté or frying pan (the largest you have — you want plenty of room and hot surface) over high heat. Drizzle in a little grapeseed or vegetable oil (not olive oil — you want an oil with a high smoke point) and heat until very hot. Add the sweet potatoes and arrange them in a single layer. Cook over high heat until they begin to caramelize and turn brown, about 4 minutes. Flip the sweet potatoes and cook for 3 minutes more.
Turn the heat down to medium and push the sweet potatoes up in a pile against one side of the pan. Add the onions to the center of the pan and season lightly with salt. Cook, stirring occasionally, until the onions are beginning to turn brown. Add the garlic and ginger and stir them into the onions. Push the onions off to the side of the pan, next to the sweet potatoes, where they will continue to caramelize.
Add the mushrooms to the hot center of the pan and cook for 4 minutes without turning them. Stir the mushrooms and cook for 4 minutes. At this point everything should be getting well-cooked; the onions should be quite dark brown and the garlic should be golden and soft. The potatoes should be softening.
Whisk together the vinegar, soy sauce, and 2 tablespoons of oil. Pour this into the pan with the vegetables and mix everything together, scraping the bottom as you go. Cook for 3 minutes.
Turn the heat up to high, as high as it will go. Add the orzo gradually, shaking in a cup at a time, and stirring and scraping constantly. Cook, letting the orzo get browned on the bottom of the pan, then scraping it up, for about 5 minutes. You are developing a little more color and flavor on the pasta, and helping all the flavors combine.
Add the greens and cook until they're barely wilted, about 1 minute. Remove from the heat, taste, and season as needed. Serve hot.