Preheat the oven to 200°C.
Kill the lobsters by freezing for thirty minutes then piercing between the eyes.
Separate the claws, tail, and head sections from the body.
Blanch the tails for 30 seconds and the claws for three minutes in boiling water, then chill in iced water.
Extract the meat from the tails and claws in whole pieces and set aside.
Heat 400ml of olive oil in a roasting tray and roast the lobster carcasses for 30 minutes.
Deglaze the roasting tray with Cognac on the hob.
Sweat chopped carrots, onions, fennel, and garlic in the remaining oil for 10 minutes.
Add spices and herbs to the vegetables and sweat for another five minutes.
Stir in tomato purée, fresh tomatoes, and 200ml of water, cooking for 10 minutes.
Combine the roasted shells and their oil with the vegetable base and cover with water.
Bring to a boil and simmer the bisque for 20 minutes.
Strain the bisque through a colander and then a fine sieve, adjusting seasoning.
Measure 200ml of bisque into a pan with butter and set aside for reheating.
Steam the carrots, fennel, leeks, and asparagus until tender and refresh in cold water.
Peel the tough skins from the broad beans.
Fry the lobster meat in oil, then heat in the oven for two minutes.
Turn the lobster meat, add chopped tarragon, and heat for two more minutes.
Reheat the buttered bisque and the steamed vegetables.
Slice the tails and arrange with claw meat and vegetables in warm bowls.
Froth the bisque with a hand blender and finish with lemon juice and seasonings.
Spoon the frothed bisque over the dish and garnish with herbs and lobster oil.