Working one lobster at a time, place on cutting board, belly side down with the head facing you. Insert a chef’s knife where the tail meets the head and quickly bisect the head lengthwise, this will quickly and easily kill the lobster. Use a cleaver or chef’s knife to cut the whole lobster lengthwise.
Heat a large 7-quart Dutch oven over medium heat, add 4 tablespoons of the olive oil and heat through. Working 1 lobster at a time, sear the lobster halves until the shells are bright red and the meat is cooked through, flipping once, about 8 to 10 minutes.
Remove the lobsters and set aside to cool, repeating with the remaining lobsters. Remove the meat from the tail and body of the lobsters and roughly chop, setting aside in a small bowl. Reserve the lobster halves for serving, and use a cleaver to crack the claws for serving.
Return the large Dutch oven to medium heat, add the olive oil and heat through. Add the butter and allow to melt then add the shallots, fennel and carrot and sauté until the vegetables are soft.
Add the garlic and sauté until fragrant then stir in the red pepper flakes followed by the white wine. Simmer the wine until the vegetables are almost dry, about 6 to 8 minutes. Add the tomato paste and tomatoes.
Cook, simmering until the sauce comes together, about 1 hour, stirring occasionally. Stir in the tarragon and cooked lobster meat.
Bring a large pot of water to the boil. Add in the pasta and cook, stirring occasionally, until al dente, about 8 to 10 minutes. Drain the pasta and set aside.
Add the cooked pasta to the sauce, tossing to combine. Divide among warmed serving bowls and arrange the reserved shells with cracked claws on the side of the bowl. Garnish with the reserved fennel fronds.