To make the sauce, put the lobster shell in a plastic bag and smash with a meal mallet or a rolling pin.
In a large frying pan over medium high heat, melt the 3 tablespoons butter. Add the lobster shell pieces and sauté, stirring frequently, until golden brown and aromatic, 2-3 minutes. Add the shallots, carrot, leek, tomato paste, bay leaf, and thyme and tarragon sprigs and reduce the heat to medium. Sauté, stirring frequently, for 2 minutes. Season with salt and white pepper.
Add the cognac and cook for 1 minutes. Add the wine and cook, stirring occasionally, until it is almost completely evaporated, about 8 minutes. Stir in the cream, 3/4 tsp salt, and 1/4 tsp white pepper. Bring to a summer, then reduce the heat to low and simmer, uncovered, stirring occasionally, until slightly thickened, about 15 minutes. Strain through a fine-mesh sieve into a bowl or measuring pitcher, pressing down on the solids to extract all of the sauce. Wipe out the pan with a paper towel to remove any bits of shell and return the sauce to the pan. Taste and adjust the seasoning, and set aside for up to 20 minutes. Reheat gently just before serving.
In a small frying pan over low heat, melt the 2 tablespoons butter. Add the lobster meat and cook, stirring occasionally, until cooked through, 5 minutes. Remove from the heat; keep warm.
Meanwhile, bring a large saucepan three-fourths full of water to a rapid boil over high heat. Add 1 tablespoon kosher salt and the pasta and cook until al dente. Drain the pasta, add it to the sauce, and toss to coat the pasta evenly. Add half of the minced tarragon and toss again. Transfer to a large warmed bowl and scatter the lobster meat and the remaining tarragon on top. Serve at once.