Place the tip of a large knife through the head of each lobster to kill them. Remove the tail and claws from the body and set aside, then remove and discard the feathery gills, stomach sac and liver. Reserve the carcasses for the sauce.
Take the lobster tail and wiggle the centre part of the narrow end of the tail until it breaks loose. Pull out the intestinal tract, using a pair of narrow pliers if it breaks, and discard. Set the tail aside.
Bring a pan of water up to a rolling boil, then plunge the lobster tails in and cook for 20 seconds. Remove with a slotted spoon and immediately refresh in iced water. Once cool enough to handle, remove the outer shell and set aside with the rest of the carcass for the sauce. Place the 4 pieces of tail meat on a cloth in the fridge.
Bring the same pan of water back to the boil and drop in the claws. Leave to cook for 5 minutes, then refresh in iced water and crack open the claws to extract the meat. Save the meat for another dish and reserve the broken claws for the sauce. Don’t throw away the water used to boil the lobster, as this will be used for the sauce as well.
To make the sauce, smash the lobster carcasses and cracked pieces of shell into small pieces by hitting them with a rolling pin. Add the olive oil and 50g of the butter to a large saucepan over a medium heat, then add the carrots and gently cook until softened but not coloured.
Add the onion and garlic and cook until softened, followed by the celery and leek. Once all the vegetables are soft, add the tomato paste, cook for 1 minute, then add the smashed lobster carcass and pieces of shell and stir to combine, frying for a few minutes.
Add the thyme and the water and bring to the boil, then reduce the heat to a simmer and cook for 15 minutes. Add the tarragon, cook for 5 minutes, then pass the liquid through a fine sieve and then a muslin cloth until you’re left with a clear liquid.
Measure out 400ml of this liquid into a pan and add the remaining 100g of butter (reserve the rest of the lobster stock for poaching the tails). Bring to the boil quickly and reduce over a high heat until it has reduced and thickened.
Meanwhile, prepare the cauliflower. Cut off 6 large florets from the cauliflower and trim them so they have 2 flat edges either side. Cut each floret in half, so you have 12 equal pieces in total which are roughly 7mm thick.
Chop the remaining cauliflower (including the stalk) and place in a pan with 100g of the butter. Add enough water to just cover the cauliflower, throw in a pinch of salt and then bring to a boil over a high heat. Cook until all of the water has evaporated, then transfer the contents of the pan into a blender and blitz until smooth (add a little extra water if you think it needs it). Pass through a fine sieve into a clean pan and set aside to reheat before serving.
Once the sauce has thickened, blitz it with a hand blender to help emulsify the butter into the stock. Keep warm over a very low heat until ready to serve, adding a little chopped black truffle a few minutes before serving.
Add the remaining 25g of butter to a frying pan and gently cook the cauliflower florets on both sides until golden, crisp and caramelised.
To cook the lobster tails, bring the reserved stock to the boil and drop in the lobster tail meat, ensuring they are fully submerged by the liquid. Poach for 4-5 minutes until cooked all the way through. Meanwhile, gently reheat the cauliflower purée.
To serve, drag a spoonful of the purée to one side of each plate, then add 3 pieces of caramelised cauliflower to the other side, interspersed with shavings of fresh truffle. Lift the lobster tails out of the stock (trimming them if needed), season with salt and then place between the cauliflower and purée. Place 2 more shavings of fresh truffle on top of each swipe of purée, then finish with the sauce and serve.