Cut the shiitake caps into bite-size pieces (quartered or halved, depending on the size).
Heat 2 tablespoons of the vegetable oil over high heat in a wok or large skillet. When almost smoking, add the mushrooms, season with salt and pepper and cook until browned and slightly crispy, 3 to 4 minutes, stirring once halfway through. Resist the urge to move them too much unless they are burning. Remove to a plate. Lower the heat to medium-high and add the remaining tablespoon oil and the Brussels sprouts. Season with salt and pepper and cook until browned or charred on one side, about 3 minutes. Stir in the ginger and cook until aromatic and slightly browned, stirring constantly, about 15 seconds. Reduce the heat to medium. Add the chicken broth (the sprouts should be just submerged in the liquid) and cook at a rapid simmer until the sprouts are tender when pierced with a knife and the liquid has boiled away, 6 to 8 minutes more.
Return the mushrooms to the wok, fold to combine and season. Serve on a platter.
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