Fillets of Sole on Beurre Rouge

Butter a noncorrosive 11 x 7.5 inch baking dish, using 1 tablespoon of the butter. Sprinkle bottom with 1 teaspoon of the shallot. Fold under ends of each fish fillet to meet; place rolled fillets 1 inch apart in prepared baking dish. Sprinkle fillets with salt and pepper. Squeeze juice from lemon half evenly over fish. Refrigerate fish, covered.

Melt 1 tablespoon of the butter in medium saucepan over medium-low heat. Add mushrooms, celery, 1 teaspoon of the shallots, the thyme, bay leaf, and peppercorns; sauté until vegetables are softened but not browned, about 10 minutes. Cut fish bones into 2-inch pieces with kitchen shears; add to saucepan. Reduce heat to very low; steam mixture, covered, 10 minutes, shaking pan occasionally. Increase heat to medium. Add 4 cups water to saucepan; heat to simmering. Adjust heat to maintain bare simmer; simmer, skimming surface occasionally, 15 minutes. Remove from heat; strain stock through sieve lined with double thickness of dampened cheesecloth. Discard solids; reserve stock.

Heat over to 350F.

Remove tender light green portion from just above whites of leeks; reserve whites for other use. Discard any tough green portions. Split leek greens lengthwise; rinse under cold running water to remove all sand and grit. Cut leek greens into strips 1/8 inch wide; blanch in small saucepan of boiling salted water until tender, about 4 minutes. Drain; rinse briefly under cold running water. Drain thoroughly on paper toweling; reserve.

Add vermouth to baking dish containing fish; pour in enough reserved fish stock to barely cover fish. Bake, uncovered, until fish is opaque in center, about 20 minutes.

While fish is baking, prepare buerre rouge. Have the remaining 1/2 cup butter cold; cut into 10 pieces. Combine red wine and the 2 tablespoons shallot in small, noncorrosive saucepan. Heat over medium heat to boiling; boil until almost all liquid is evaporated, about 5 minutes, reducing heat during last few minutes to prevent scorching. Remove pan from heat; whisk in 1 piece of butter. Return pan to low heat; continue whisking in remaining butter, one piece at a time, adding next piece of butter before butter in saucepan in entirely incorporated. (Sauce should be creamy and of half-melted consistency; if butter begins to melt too quickly and becomes oily, remove pan from heat immediately and whisk in next 2 pieces of butter.) Stir in reserved leek greens; season sauce to taste with salt and pepper.

Pour a pool of sauce into each of 4 warmed small plates, dividing evenly. Remove sole from poaching liquid with slotted spatula, draining thoroughly; place one rolled fillet over sauce on plates. Server immediately.