Fit electric mixer with a dough hook. In the bowl of the mixer, combine ricotta, flour, 1 tablespoon oil, and salt. Form a well in the middle of the mixture and crack yolks into middle of the well. Using the dough hook, mix until the ingredients just come together. Wrap dough in moist towel and let rest in a warm location for 10 minutes. Roll dough through a pasta machine starting at the largest setting and slowly working to the number 4 setting. Take the sheet of dough and cut into strips about 1/2-inches wide by 10 inches long, then pass through cavatelli machine. Place the pasta on sheet tray and freeze until ready to use.
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, chop 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 chili 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.