Preheat the oven to 425 degrees F.
To make the mushrooms, toss caps with olive oil, ginger, and salt, to taste. Place the mushrooms in a roasting pan and roast in the oven for 5 minutes, stirring occasionally. Remove the mushrooms from pan, cool, add soy sauce, and set aside.
In a medium size mixing bowl, combine the root beer, hoisin, vinegar, mirin, and shallots. Using a wire whisk, beat while slowly adding the oil a thin stream. Set aside.
To prepare the fish, preheat a saute pan and add the peanut oil. Season the fish with salt and pepper and place in the pan. Saute for 3 to 4 minutes per side or until fillet just becomes opaque. Set aside and keep warm.
To serve, place fish in the center of each plate. Top with 4 mushroom caps each and drizzle the vinaigrette over the fish.