In large sauce pan, bring enough water to cover lobster to a boil. Plunge lobster into pot head first.
Reheat to boiling; reduce heat to medium, cover and simmer 12-15 minutes until shell is red, lifting lid occasionally to let steam escape.
Place boiled lobster aside to cool until it can be handled. Remove meat. Reserve the coral and the green fat.
Put the cold water into a sauce pan and add the broken claws and shells. Bring to the simmering point and simmer 20 minutes.
Melt butter in 2 quart saucepan over medium heat. Add milk until warmed.
Slowly whisk in flour until a sauce is formed.
Drain the lobster broth and add to the sauce. Dice the lobster meat and add.
Season with salt, pepper and cayenne pepper to taste. Just before serving, add the coral mashed with the green fat. Mix thoroughly, reheat and serve.