Cut the lobster down the center of the head between the eyes.
Leave the tail whole.
Separate the tail, the head, and remove the claws.
Keep any juices, the tomalley and the roe in a bowl and refrigerate.
Heat a 12 inch saute pan on high heat and add the oil.
Sear the cut pieces of lobster cut side down, and add the tail and claws whole.
Cook until the shells are bright red and the flesh is almost cooked, about 8 minutes.
Remove lobster from the pan and set to the side.
Add 2 tbsp butter to the pan and saute the shallots, carrot, celery and garlic until lightly browned.
Add tomato paste and saute for 2 minutes.
Add white wine, cook for 3 minutes, whisking to evenly disperse the tomato.
Add fish stock.
Add bouquet garni and lobster to the pot and bring to a boil, skimming any froth or fat with a 4 oz ladle.
Season with salt, pepper and cayenne, and simmer for 15 minutes.
Remove lobster and set aside, remove garni and discard.
Melt remaining 4 tbsp of butter in a small saucepan, add flour and whisk to make the roux.
Add reserved lobster juices, tomalley, roe and chopped tarragon, and whisk until smooth.
Add the cognac to the roux, and then 1/3 at a time, add the roux to the bisque, whisking to remove any lumps before each addition.
Simmer for 15 minutes.
Strain through a china cap, using a ladle or rubber spatula to force as much as possible through.
Add cream and season with salt and pepper.
Remove meat from lobster and cut into bite size chunks.
Serve with lobster meat, a dollop of creme fraiche and a sprig of fresh tarragon.