Melt 4 tbsp of the butter in a wide saucepan and, when foaming, add the onion, carrot, and garlic. Sauté until golden but not brown. Add the bouquet garni, and season with salt and pepper. Place the eel pieces on top and pour the wine over - there should be sufficient just to cover the fish. Bring to a boil and simmer for 20-25 minutes.
Meanwhile, cook the pearl onions in boiling salted water until tender. Drain and keep hot. Sauté the mushrooms in the remaining butter. Keep hot.
Remove the eel with a slotted spoon and place in a serving dish with the pearl onions and mushrooms. Strain the cooking liquid into a clean saucepan and, if necessary, reduce by rapid boiling. Then simmer, gradually adding the beurre manié in small pieces, whisking well between each addition, to thicken the sauce slightly. Pour the sauce over the eel and serve garnished with fried croûtons.
© 1999 Jennifer Paterson and Clarissa Dickson Wright estate. All rights reserved.