Remove the meat from the shell. Cover and cool.
Melt the butter in an enameled cast iron saucepan or a flame-ware saucepan. Add the lobster meat and simmer over low heat for a few minutes.
Add the milk and stir most of the time until hot.
Add the cream and keep stirring. When hot, simmer for a few minutes, but do not boil. Cool.
Cover and refrigerate 12 to 24 hours before serving to develop the full flavor of the lobster.
Reheat without boiling, and just before serving, season to taste with salt and pepper.