1. Pour one recipe's worth of Mushroom Essence through a fine strainer into a 4- to 5-quart pan; discard the ginger. Put mushrooms in a shallow 10- to 11-inch oval casserole (about 2 1/2 qt.). Add 1/2 cup green onions, capers, sherry, soy sauce, and ginger to mushrooms; mix and spread level. Lay sole fillets evenly over mixture, overlapping if necessary. Cover casserole tightly with foil.
2. Bake sole in a 375º oven until fillets are opaque but still moist-looking in center of thickest part, about 15 minutes.
3. While sole bakes, add potatoes and tarragon to mushroom broth. Bring to a boil over high heat; cover and simmer until potatoes are soft enough to mash easily, about 15 minutes. Drain potatoes in a strainer over a bowl. Pour liquid back into pan and put potatoes in the bowl.
4. Uncover sole and drain juices into pan. Boil liquid over high heat until reduced to about 3/4 cup, 4 to 5 minutes. In a small bowl, blend cornstarch with 1 tablespoon water and stir into boiling liquid; stir until boiling again, and remove from heat. Turn oven to broil.
5. Meanwhile, with a potato masher or a mixer, mash potatoes until smooth, adding enough milk for desired texture. Push sole and mushrooms toward center of casserole to leave a 1-inch space around the edge. Spoon potatoes (or pipe through a pastry bag with a large plain or star tip) into space. Pour sauce over sole mixture. Sprinkle potatoes with cheese.
6. Broil about 4 inches from heat until cheese is very lightly browned, about 4 minutes. Sprinkle with remaining 1/4 cup green onions. Add salt and pepper to taste.