Soak the dried mushrooms in 3 cups very hot water until soft, anywhere from 5 to 15 minutes. Dried shiitake are much tougher than other varieties and should be soaked in boiled water. When they are tender, remove the mushrooms from the liquid with a slotted spoon, reserving the liquid and discarding the stems; slice or chop if the pieces are large.
Meanwhile, set a pot of water to boil for the broccoli. Cook the broccoli for 2 minutes in the boiling water, then drain.
Put a large, deep skillet over medium-high heat; add the oil and swirl it around, then add the garlic and ginger. Cook for 15 seconds; add the onion and cook, stirring occasionally, until beginning to soften and brown, 3 to 5 minutes. Add the fresh and soaked dried mushrooms, and allow them to cook down 2 or 3 minutes before adding the carrots and celery. Cook, stirring occasionally, until vegetables are tender but not at all mushy, 10 to 12 minutes. Add the broccoli after about 6 minutes of cooking.
If you’re using the cornstarch, dissolve it in the soy sauce; stir into the pan and sprinkle with salt and pepper. Add the crushed red chili flakes if you’re using them, and pour in 3/4-cup of the mushroom-soaking liquid. Stir the mixture, and scrape the bottom of the pan, then turn off the heat; the liquid should be mostly absorbed. Sprinkle with the scallions.