bring 2 large pots of water to boil and salt heavily (salty like the sea). while water is coming to boil, pass tomatoes through a food mill and then chinoise to remove seeds. set aside. thinly slice garlic and set aside. remove the tail and claws of the lobsters and place the claws in 1 of the pots of boiling water. turn the heat off and wait 2 minutes. add the tails. let steep 6 minutes more and remove tails and claws to ice water. when cooled, remove meat, cutlet and reserve. add cavatelli to other pot and cook about 8 minutes.
in saute pan, heat 4 ounces oil and add garlic, cooking 1 minute until garlic starts to just brown. add sliced chilli and cook 1 more minute. stir in 1/4 cup dry white wine, cooking 1 minute, then add tomato puree. cook 3 minutes more. when pasta is cooked, add it to the sauce with the lobster and cook 2 minutes. add butter and turn off heat. stir and season, to taste, with sea salt and white pepper.
to serve, divide the pasta among 4 bowls, tear the basil leaves with your hand and sprinkle over the top of the pasta. drizzle with extra-virgin olive oil.
this recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. the food network kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.