Preheat the water bath to 131°F (55°C).
Place the butter, chili paste, crushed garlic and parsley into a vacuum sealer bag.
Season the lobster with salt, then add the lobster to the bag and vacuum seal. Lower the bag into the pot and allow it to cook for 1 hour while you make the sauce and pasta.
For the sauce, heat the oil in a heavy-bottomed large saucepan over high heat. Add the reserved lobster shells and sauté the shells for about 3 minutes. Reduce heat to low, stir in the tomato paste and continue stirring for 5 more minutes.
Add the tomatoes, wine, vinegar, sliced garlic, thyme, parsley, salt and pepper and simmer for 4 to 5 minutes until the alcohol from the wine cooks out.
Add the cream and bring to a boil, then reduce the heat to medium-low.
Stirring occasionally, simmer the sauce just until lobster flavor infuses the cream and the sauce slightly thickens, about 20 minutes.
Strain the sauce into a fresh saucepan with cheesecloth or a sieve, pressing on the solids to extract as much liquid as possible, then put the saucepan over low heat.
Make any seasoning adjustments and simmer 5 to 10 minutes for sauce to thicken. Remove the pan from the heat.
Cook and drain the pasta per directions on the box.
Remove the lobster from the water bath and empty the vacuum sealer bag, saving only the lobster and 2 tablespoons (30 ml) of the liquid, discarding any herbs left in the bag.
Cut the tails into small bite-size pieces and add to the sauce along with the cooking liquid, tossing to coat.
Add the cooked pasta to the saucepan. Toss it all together until completely covered in sauce and serve garnished with basil leaves.