1. In a large pot, bring about 1 inch of water to a boil. Sprinkle the fish with 1/2 teaspoon of the salt and 1/4 teaspoon of the pepper. Fold each fillet in half and put the fillets in a steamer basket over the boiling water. Reduce the heat to moderately high, cover, and steam until just done, about 10 minutes for folded 1/2-inch-thick fillets. Transfer the fillets to plates.
2. Meanwhile, in a large frying pan, heat the 2 tablespoons oil over high heat. Add the mushrooms and the remaining 1/4 teaspoon each of salt and pepper. Cook until browned, about 5 minutes. Top the fish with the mushrooms.
3. Reduce the heat to moderately low. Heat the remaining teaspoon of oil in the pan; add the ginger, scallions, and garlic. Cook, stirring, for 1 minute. Stir in the soy sauce, dry mustard, wine, and broth. Simmer, stirring occasionally, until slightly thickened, about 2 minutes. Spoon the sauce over the mushrooms and fish.
Fish Alternatives: Steam fillets of sole, whitefish, or turbot to pair with the mushroom sauce. If the fillets are thin, either fold them in half or roll them up before steaming.
Wine Recommendation: An acidic, medium-bodied white wine is the best choice to pair with the spicy, salty Asian flavors in this dish. Try a wine made from a Loire Valley chenin blanc, such as Vouvray