1. In a heavy 3- to 4-quart saucepan combine the eel, quartered onion, bay leaves and parsley sprigs, and cover with water. Bring to a boil over high heat, reduce the heat to low, partially cover the pan, and simmer for 20 minutes, or until the eel is firm to the touch. Do not overcook.
2. With a slotted spoon, transfer the eel to a plate and strain the stock through a fine sieve into a bowl. Discard the vegetables. When the eel is cool enough to handle, remove the skin and cut the flesh away from the bones. Spread the pieces side by side in a 6-by-6-by-2-inch casserole or baking dish and set aside.
3. Preheat the oven to 450°. In a heavy 8- to 10-inch skillet, melt 1 tablespoon of the butter over moderate heat. When the foam begins to subside, add the sliced onion and cook, stirring frequently, for 5 minutes, or until the slices are soft and transparent but not brown. Add the sherry, chopped parsley, nutmeg, cayenne pepper and salt, and set aside off the heat.
4. In another 8- to 10-inch skillet, melt the remaining 3 tablespoons of butter over moderate heat. When the foam begins to subside, stir in the flour and mix thoroughly. Pour in 2 cups of the reserved eel stock and, stirring constantly with a whisk, cook over high heat until the sauce thickens heavily and comes to a boil.
5. Reduce the heat to low and simmer for about 3 minutes to remove any taste of raw flour. Then add the lemon juice and the reserved onion-sherry mixture. Taste for seasoning. Pour the sauce over the eel slices, arrange the quartered eggs on top, and let the sauce cool to room temperature.
6. On a lightly floured surface, roll the puff pastry into a rough square about ¼ inch thick. Drape the pastry over the rolling pin, lift it up and unfold it over the baking dish.
7. Trim off the excess with a small, sharp knife and, with the tines of a fork or your fingers, crimp the pastry to secure it to the rim of the dish. Gather the scraps of pastry into a ball, reroll and cut them into simple leaf and flower shapes.
8. Moisten their bottom sides lightly with the egg yolk- and-cream mixture, and arrange them decoratively on the pie. Then brush the entire pastry surface with the remaining egg-yolk-and-cream mixture and cut 3 parallel 1-inches slits ½ inch apart in the top.
9. Bake the pie in the middle of the oven for 15 minutes, then reduce the heat to 325° and bake for 30 minutes longer, or until the top is golden brown. Serve at once.
