Prepare Basic Tomato Sauce according to recipe and set aside, keeping sauce at low simmer.
To prepare the artichokes, in a 12 to 14-inch saute pan, combine artichokes, 1 cup oil, mint, chili flakes and 2 thinly-sliced cloves of garlic and place over medium heat until center of pan has reached full simmer. Artichokes should be submerged in oil. Remove from heat and let cool 15 to 20 minutes. Set aside.
To prepare the sweet garlic, combine 2 tablespoons oil and remaining garlic in a 12 to 14-inch saute pan and cook over high heat until cloves are browned. Add Cinzano and reduce liquid until syrupy. Set aside.
Bring 6 quarts of water to a boil and add 2 tablespoons salt.
To prepare the lobster, fill a pot large enough to hold all lobsters 3/4 full with water and bring to a boil. Steam lobsters in water until done, approximately 12 minutes. Remove from heat, let cool slightly and remove all meat from claw and tail.
(All of preceding components can be prepared 8 hours in advance and refrigerated.)
In a large saute pan, combine tomato sauce, drained artichokes, sweet garlic and lobster meat and place over medium heat to just warm through. Season with salt and pepper to taste.
Cook pasta according to package directions, until tender yet all dente, and drain. Add drained pasta and toss over medium heat to coat, about 30 seconds. Divide into 4 heated bowls and serve immediately.
Wine Recommendation: Marco Felluga, Pinot Grigio, Italy, 1998
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds one week in the refrigerator or up to six months in the freezer.