Preparation

• Bring a large pot of salted water to a boil.
• Kill the lobsters quickly by holding each one to face you, then driving a knife between the eyes and pulling it down through the head, like a lever. (Watch a video of this technique.)Using a heavy knife, split the lobster heads in half down the middle, then remove the heads, chop each one into 6 or 8 small pieces, and set the pieces aside. Separate the claws and tails from the bodies by twisting them, and discard the bodies or freeze them for another use. Put the claws into the boiling water and cook for 1 minute, then add the tails and cook for 1 minute more. Use a slotted spoon to remove them from the water and place them on a clean surface to cool. (You can put them in the refrigerator to speed the process if you like.) The lobster meat will appear quite underdone, but it will finish cooking later. Once it is cool, shell the lobster, splitting the tails and keeping the claw meat as intact as possible. Wrap the lobster meat in plastic wrap and refrigerate while you finish making the dish. Reserve the shells for the sauce.
• Heat the oil in a large heavy-bottomed saucepan over medium-high heat until it is hot but not smoking. Add the lobster head pieces and reserved shells and cook, stirring a few times, for 5 minutes. (You should smell a potent oceanic aroma as the flavor is released.) Add the tomato paste, if using, and curry and stir for another 2 minutes. Add the carrots, onions, garlic, and ginger and cook, stirring, for another 10 minutes. Remove from the heat and pour in the Cognac and Sauternes. Return to high heat and reduce by about two-thirds, until about 1 cup remains, 10 to 15 minutes. Pour in the stock and bring back to a boil. Reduce the heat to a simmer and cook for 1 hour. Season to taste with salt and pepper.
• Strain the broth through a fine-mesh strainer set over a large bowl. Wipe out the pot and empty and rinse out the strainer. Line the strainer with cheesecloth, set it over the pot, and strain the broth again, back into the pot. Stir in the cream and bring to a boil, keeping a close eye on the pot so it doesn’t boil over. Reduce the heat to medium and cook until slightly thickened and flavorful, about 30 minutes. Season to taste with lime juice and salt (keeping in mind that lobster is naturally salty). Remove any extra bits of lobster shell by straining the sauce through a fine-mesh strainer or cheesecloth set over a medium saucepan. Set the pan over medium heat and whisk in the butter. Add the lobster meat and reheat gently for 2 to 3 minutes.
• To serve, place 2 claws and 1 whole tail in each of 4 shallow bowls, then spoon some sauce over each serving. Sprinkle with the basil and serve immediately.