Bring the wine and 3 bottles of clam juice to a boil over high heat in a stockpot large enough to hold the lobsters. Rinse the lobsters under cold, running water and put them in the pot. Cover the pot and steam the lobsters for 3 minutes. Shake the pot to redistribute the lobsters and steam for 4 minutes more. Remove the cover, being careful to avoid steam as it escapes. Use large tongs to transfer the lobsters to a large bowl. Strain the liquid through a fine-mesh strainer and set aside.
When the lobsters are cool enough to handle, remove meat from the tail and claws and dice. Place the meat in a bowl, cover with plastic wrap, and refrigerate. Split the lobster body and head in half lengthwise with a heavy knife and remove inedible portions.
Grind the shells in a food processor in 2 or 3 batches. (A small amount of thin pinkish paste will adhere to some of the shells.) Heat a 12-inch heavy-bottomed skillet over high heat until very hot, about 3 minutes. Add 1 tablespoon olive oil and swirl to coat the pan bottom. Add half of lobster shells; sauté until aromatic and lightly browned, about 3 minutes. Transfer the browned shells to a medium bowl and repeat with the remaining oil and shells. Return the first batch of browned shells to skillet. Pour the warmed brandy over the lobster shells and flambé, shaking the pan. When the flames subside, turn off the heat and hold the shells in the pan.
Meanwhile, heat 2 tablespoons butter in a large Dutch oven over medium heat until foaming. Add the carrot, celery, onion, and garlic and cook, stirring frequently, until the vegetables are slightly softened and lightly browned, 6 to 7 minutes. Add the remaining 4 tablespoons of butter and stir with a wooden spoon until melted. Add the flour and cook, stirring constantly, until combined thoroughly, about 1 minute. Slowly stir in reserved liquid from steaming lobsters, initially adding about 1/4 cup at a time. Stir until blended, then continue adding the remaining liquid and scraping the pan bottom with a wooden spoon to loosen browned bits. Add the tomatoes, tarragon sprig, and lobster shells. Pour remaining bottle of clam juice into the skillet once shells have been removed. Turn heat to high and bring liquid to a boil, scraping up browned bits from the pan bottom. Add the clam juice mixture to the pot with the shells. Once the liquid in the Dutch oven comes to a boil, cover the pot, reduce the heat to low, and simmer, stirring frequently, until thickened, about 20 minutes. Stir in the cream and simmer for 10 minutes longer.
Strain the bisque through a chinois or a fine-mesh strainer into a medium container, pressing on the solids with the back of a ladle to extract all liquid. Wash and dry the now-empty Dutch oven. Return the strained bisque to the Dutch oven and stir in the lemon juice and cayenne. Bring to a simmer over medium-high heat. When piping hot, add the diced lobster meat (add claws first if serving whole, cook 1 minute, then add remaining meat) and sherry and then season to taste with salt and pepper. Serve immediately, garnishing each bowl with a little chopped tarragon.