Bring a large pot of generously salted water to a boil. Add the lobster tails and cook for 5 to 6 minutes, or until the shells are bright red and the meat is just opaque. Remove the lobster tails and set aside until cool enough to handle.

Return the same pot of water to a boil. Cook the pasta according to the package directions until al dente, reserving 1/2 cup of the pasta water before draining.

While the pasta cooks, remove the lobster meat from the shells and cut into bite-sized pieces.

Heat the olive oil in a large skillet over medium heat. Add the shallot and cook for 2 to 3 minutes, until softened. Season lightly with a pinch of kosher salt. Add the garlic and cook for 30 seconds, just until fragrant.

Stir in the Calabrian chili paste, then pour in the white wine. Simmer for 3 to 4 minutes, allowing the wine to reduce by about half.

Add the cherry tomatoes, gently breaking them apart with the back of a spoon. Simmer for 6 to 8 minutes, stirring occasionally, until the sauce thickens slightly. Season with kosher salt and a few grinds of freshly ground black pepper. Reduce the heat to low and stir in the heavy whipping cream until smooth and combined.

Add the lobster meat and cook for 1 to 2 minutes, just until warmed through. Add the drained pasta and toss until evenly coated, adding a splash of the reserved pasta water as needed until the sauce is silky and coats the pasta.

Remove the skillet from the heat and stir in the basil and tarragon. Taste and season with additional kosher salt and freshly ground black pepper, if needed. Serve immediately.