I found the basics of this recipe in an old cookbook years ago and have been evolving it ever since. It is a bit of a production but well worth the effort especially on special occasions like New Year's Eve, last night. Guests say it is the best lobster they ever ate.
I use lobster stock for the base of the sauce but, fish or shrimp stock will do. The lobster stock does intensify the flavor.
I serve it with a salad and crusty bread to soak up the sauce, but it will do well over pasta or even rice. —Stephen Gray
Cut up the lobsters into claws, knuckles and split tails. I do this live, but if you are squeamish, you can plunge the lobsters into boiling water for 2 minutes to kill them first then cool in ice water. (I think that boiling them first does dilute the lobster flavor of the dish.)
Make stock from the bodies for another day or discard. (Bodies can be cooked witht the other pieces but you will need a pig pan.)
Crack the claws for easier eating.
Partially cook the lobster pieces in olive oil over medium high heat (about 6-7 minutes for the tails and a little longer for the claws. Set aside
(Note: As you saute the lobster a white fluid will be exuded into the pan. Remove this as it appears or it will burn in the pan and become bitter.)
In the same pan (after you remove any burned white bits), sauté garlic, shallots, celeriac, carrots in olive oil till slightly softened. Add lobster stock and tomatoes and cook for a couple of minutes longer.
Add Cognac and flame. Add wine, tarragon, salt, pepper and stir in butter.
Add lobster to the pan and simmer till the lobster is fully cooked and the sauces is a bit reduced.
Place the lobster on a plate, garnish with parsley and serve with crusty bread