For the bisque, cut the cooked lobsters in half lengthways and remove the meat. Roughly chop the lobster meat, except the claw meat, into 2cm/¾in pieces. Reserve the claw meat and lobster shells, and keep everything in the fridge until needed.
Heat half of the butter in a frying pan until foaming, add the onion, carrot and thyme and cook for 2-3 minutes, or until softened but not coloured.
Add the reserved lobster shells and tomato purée and cook for a further minute, stirring well.
Add the white wine and cook until reduced in volume by half.
Add the fish stock, bring the mixture to the boil, then reduce the heat and simmer for 10 minutes, until the liquid has reduced and thickened a little.
Take off the heat, and allow to cool slightly, then blend in batches in a food processor to a purée.
Strain the puréed bisque through a sieve into a clean saucepan and whisk in the remaining butter.
Season, to taste, with salt and freshly ground black pepper. Keep warm.
For the spaghetti, bring a large pan of salted water to the boil, add the spaghetti and cook according to the packet instructions.
Drain the spaghetti, reserving a little of the cooking water, and set aside.
Meanwhile, heat the olive oil in a frying pan and add the shallots and garlic and cook for two minutes.
Add the chopped lobster meat and chopped tomato and heat through.
Season with salt and freshly ground black pepper then add the basil, spaghetti, reserved cooking water and a couple of ladles of the bisque. Toss together. (Save the remaining lobster bisque for use in other dishes.)
Heat a frying pan until hot, add the butter and reserved lobster claw meat and cook for 1-2 minutes on each side.
To serve, pile the spaghetti into serving bowls and top with one lobster claw each and a few basil leaves.