1. In a heavy 3- to 4-quart saucepan, melt the butter over moderate heat. When the foam begins to subside, stir in the flour and mix thoroughly.
2. Pour in about ½ cup of the fish stock (or degreased chicken stock) and, stirring constantly with a whisk, cook over high heat until the sauce thickens and comes to a boil.
3. Reduce the heat to low and simmer for about 3 minutes to remove any taste of raw flour. Stir in the eggs, then add the remaining 2½ cups of stock, the cream, milk, salt and pepper.
4. Still stirring, bring to a boil over high heat. Reduce the heat to moderate and add the crab meat, sherry, lime juice and bitters. Cook for 2 or 3 minutes to heat the crab meat through.
5. Taste for seasoning and serve at once from a heated tureen or in individual soup plates.
