Make broth: Wash chopped leek well in a bowl of cold water, agitating it, then lift out and pat dry.
Heat oil in a 3- to 4-quart heavy saucepan over moderately high heat until hot but not smoking, then sauté leek, carrot, onion, and garlic clove, stirring frequently, until vegetables are soft and well browned, about 10 minutes. Stir in wine and deglaze saucepan by boiling, stirring and scraping up any brown bits, 1 minute.
Add water, soy sauce, parsley, thyme, bay leaf, and porcini and simmer, uncovered, until liquid is reduced to 3/4 cup, about 25 minutes. Pour through a fine-mesh sieve into a glass measure, lightly pressing on and then discarding solids. Transfer to a small saucepan. If you have more than 3/4 cup, boil strained liquid a few minutes to reduce.
Sauté fresh mushrooms: Heat butter and oil in a 10-inch heavy skillet until hot but not smoking, then sauté mushrooms with salt and pepper, stirring, until just tender and golden brown, about 4 minutes. Add celery and sauté until bright green and crisp-tender, about 2 minutes. Remove from heat and keep warm, covered with foil.
Finish sauce and sauté skate: Bring broth to a simmer, then remove from heat and stir in tapioca. Let stand, covered, 10 to 15 minutes.
While sauce is standing, stir together flour and curry powder in a shallow bowl. Pat fish dry and sprinkle with salt and pepper, then dredge in flour mixture, shaking off excess and transferring to a plate as dredged.
Heat butter in a 12-inch heavy skillet over moderately high heat until foam subsides, then sauté fish, turning over once, until golden brown and just cooked through, about 5 minutes total.
Reheat sauce, then season with salt and pepper. Serve fish with sautéed mushrooms and sauce.
*Available at specialty foods shops.
Cooks' note: Broth (without tapioca) can be made 2 days ahead and cooled completely, uncovered, then chilled, covered.